Overview, reasonable judgement
As with the frequently used standard, the "reasonable person", whether an assembly is "peaceable" or not is a question of judgement. In some cases it is clear. A group throwing stones and Molotov Cocktails at a group of opponents, or at the police, is not acting "peaceably". What about a group holding signs and chanting "No Justice, no Peace"? Calmly singing 'We Shall Overcome"?
In short, there is no bright-line separating "peaceable" from not. It is not like a trespassing case where one either did or did not cross a boundary, or a speeding case where one did or did not go faster than the limit.
There are, however, an number of fact patterns in which assembled protesters have been held not to have justified arrest, and various other circumstances in which dispersal and arrests have been held justified. See the cases below.
Each person must use judgement as to whether a group's actions are "peaceable" when deciding whether to join or remain a part of such a group. A police official must make a similar judgement when ordering a group to disperse. Later, a court may have to make a similar judgement when deciding whether someone accused of "riot" or "disorderly conduct" was within his or her constitutional rights, or if a police officer acted unconstitutionally. The circumstances and the details of the actions will matter.
The right of assembly, like the right of speech is subject to "time, place, and manner" restrictions, so long as such restrictions are content-neutral. See this page from teh Library of Congress
To be clear, a police officer's judgement does not make a group's actions constitutional or criminal, that decision is for a court. But an officer must make an on-the-spot judgment as to whether, and if so how, to react to the group's actions. If the court later finds the officer's actions reasonable in light of the specific events and circumstances, it may treat the officer differently than if it finds the LEO actions unreasonable. But my point above was that everyone involved had to make judgements, and that the law expects everyone to act in a "reasonable" manner, and there may be consequences later for those held not to have done so.
Selected Cases
Shuttlesworth
In Shuttlesworth v. City of Birmingham, 394 U.S. 147 (1969) The US Supreme Court dealt with a case involvign a civil rights protest march. Justice Stewart described the events as follows in the majority opinion:
On the afternoon of April 12, Good Friday, 1963, 52 people, all Negroes, were led out of a Birmingham church by three Negro ministers, one of whom was the petitioner, Fred L. Shuttlesworth. They walked in orderly fashion, two abreast for the most part, for four blocks. The purpose of their march was to protest the alleged denial of civil rights to Negroes in the city of Birmingham. The marchers stayed on the sidewalks except at street intersections, and they did not interfere with other pedestrians. No automobiles were obstructed, nor were traffic signals disobeyed. The petitioner was with the group for at least part of this time, walking alongside the others, and once moving from the front to the rear. As the marchers moved along, a crowd of spectators fell in behind them at a distance. The spectators at some points spilled out into the street, but the street was not blocked and vehicles were not obstructed. At the end of four blocks, the marchers were stopped by the Birmingham police, and were arrested for violating § 1159 of the General Code of Birmingham.
The opinion went on to say:
There can be no doubt that the Birmingham ordinance, as it was written, conferred upon the City Commission virtually unbridled and absolute power to prohibit any "parade," "procession," [Footnote 1] or "demonstration" on the city's streets or public ways. ... This ordinance as it was written, therefore, fell squarely within the ambit of the many decisions of this Court over the last 30 years, holding that a law subjecting the exercise of First Amendment freedoms to the prior restraint of a license, without narrow, objective, and definite standards to guide the licensing authority, is unconstitutional. ...
It is argued, however, that what was involved here was not "pure speech," but the use of public streets and sidewalks, over which a municipality must rightfully exercise a great deal of control in the interest of traffic regulation and public safety. That, of course, is true. We have emphasized before this that "the First and Fourteenth Amendments [do not] afford the same kind of freedom to those who would communicate ideas by conduct such as patrolling, marching, and picketing on streets and highways, as these amendments afford to those who communicate ideas by pure speech." Cox v. Louisiana, 379 U. S. 536 ... But our decisions have also made clear that picketing and parading may nonetheless constitute methods of expression, entitled to First Amendment protection. Cox v. Louisiana, supra; Edwards v. South Carolina, 372 U. S. 229; Thornhill v. Alabama, 310 U. S. 88.
The Court overturned the conviction of Shuttlesworth. It did not deal with the peacefulness of the march, except in the above quoted description of events, no allegation of violence was made, only of lack of a permit (which had been applied for and refused, with an indication that no permit would ever be granted to that group.)
Ward
In Ward v. Rock Against Racism, 491 U.S. 781 (1989) The Copurt held that:
even in a public forum, the government may impose reasonable restrictions on the time, place, or manner of protected speech, provided the restrictions
are justified without reference to the content of the regulated speech, that they are narrowly tailored to serve a significant governmental interest, and that they leave open ample alternative channels for communication of the information. (Clark v. Community for Creative NonViolence, 468 U. S. 288)
Lest any confusion on the point remain, we reaffirm today that a regulation of the time, place, or manner of protected speech must be narrowly tailored to serve the government's legitimate, content-neutral interests, but that it need not be the least restrictive or least intrusive means of doing so.
Ward was a Speech (music) case, not an assembly case, but the standard from Ward has also been applied in assembly cases.
Jones
Jones v. Parmley, 465 F.3d 46, 56–57 (2d Cir. 2006) was a right-of-assembly case. In it, people were accused of violating the very law (NYS 240.10) mentioned in the question, as well as other NY State laws, including disorderly conduct. The events described produced a whole series of court decisions. The linked opinion is from the US 2nd Circuit Court of Appeals, not the Supreme Court, but it reviews and quotes many relevant SC rulings. The linked opinion is actually on a claim by various law-enforcement officers (LEOs) for qualified immunity in a suit by people arrested. When it was brought, the various charges against those arrested had already been dismissed or found to be without merit.
The Court of Appeals opinion says, in relevant part:
The Supreme Court has declared that the First Amendment protects political demonstrations and protests — activities at the heart of what the Bill of Rights was designed to safeguard. See Boos v. Barry, 485 U.S. 312 ... Indeed, the Court has repeatedly held that police may not interfere with orderly, nonviolent protests merely because they disagree with the content of the speech or because they simply fear possible disorder. See Cox v. Louisiana, 379 U.S. 536 ...
First Amendment protections, while broad, are not absolute. Regan v. Boogertman, 984 F.2d 577, 579 (2d Cir. 1993) It is axiomatic, for instance, that government officials may stop or disperse public demonstrations or protests where "clear and present danger of riot, disorder, interference with traffic upon the public streets, or other immediate threat to public safety, peace, or order, appears." Cantwell v. Connecticut, 310 U.S. 296, (1940). Indeed, where a public gathering threatened to escalate into racial violence and members of a hostile crowd began voicing physical threats, the Supreme Court expressly sanctioned police action that ended the demonstration and arrested the speaker, who defied police orders to cease and desist. Feiner v. New York, 340 U.S. 315 ... the Supreme Court has long applied the "clear and present danger" test to protest cases to determine when police interference is constitutional. Moreover, although defendants make much of the fact that some demonstrators had allegedly violated the law, transforming the peaceful demonstration into a potentially disruptive one, the Supreme Court has expressly held that " [t]he right to associate does not lose all constitutional protection merely because some members of the group may have participated in conduct or advocated doctrine that itself is not protected." NAACP v. Claiborne Hardware Co., 458 U.S. 886 ...
The Supreme Court has already ... made clear that the police may not interfere with demonstrations unless there is a "clear and present danger" of riot, imminent violence, interference with traffic or other immediate threat to public safety. Cantwell, 310 U.S. at 308-309 (finding no imminent violence where anti-Catholic diatribe angered listener and provoked suggestion of violence). Neither energetic, even raucous, protesters who annoy or anger audiences, nor demonstrations that slow traffic or inconvenience pedestrians, justify police stopping or interrupting a public protest. Cox I, 379 U.S. at 546-47, 549 n. 12 (group of protesters who provoked a visceral, angered response and slowed traffic did not jeopardize their speech rights); Edwards, 372 U.S. at 232, 237 ("clear and present danger" means more than annoyance, inviting dispute or slowing traffic). ... Plaintiffs also allege that they made no threats of physical harm to police or members of the public, did not incite violence or disorder and displayed no dangerous weapons. ...
We are mindful that the First Amendment does not insulate individuals from criminal sanction merely because they are simultaneously engaged in expressive activity.
Section 240.10 of the [NY] Penal Law states that four or more persons assembled for purposes of engaging in violent and tumultuous conduct likely to cause public alarm constitutes an unlawful assemblage. N.Y. Penal Law § 240.10. Conviction under this law requires "an incitement which is both directed towards and likely to produce imminent violent and tumultuous conduct." Jones, 2005 WL 928667 ...
New York courts have interpreted [N.Y. Penal Law § 240.20(5) (disorderly conduct)] to permit punishment only where the conduct at issue does more than merely inconvenience pedestrian or vehicular traffic. People v. Pearl, 66 Misc.2d 502, 321 N.Y.S.2d 986, 987 (1st Dep't 1971) ("Something more than the temporary inconvenience caused to pedestrians by the demonstrators' blocking of the west crosswalk, requiring them to enter the roadway to get to the other side, was required to sustain a conviction for obstructing pedestrian traffic.");
[P]laintiffs had an undeniable right to continue their peaceable protest activities, even when some in the demonstration might have transgressed the law. Claiborne Hardware, 458 U.S. at 908. Plaintiffs still enjoyed First Amendment protection, and absent imminent harm, the troopers could not simply disperse them without giving fair warning. *City of Chicago v. Morales-, 527 U.S. 41, 58 ("[T]he purpose of the fair notice requirement [in disorderly conduct statutes] is to enable the ordinary citizen to conform his or her conduct to the law."); Feiner, 340 U.S. at 321, 71 S. Ct. 303 (finding no First Amendment violation where imminence of disorder was "coupled with petitioner's deliberate defiance of the police" and their orders to disperse)
Specific actions
Actions that have been held to be permitted as part of a protest or assembly, not justifying police intervention, include:
walking in orderly fashion, remaining on the sidewalks except at street intersections, not interfering with other pedestrians or traffic;
picketing and parading, even without; a permit when a permit has been improperly denied;
orderly, nonviolent protests;
A diatribe that angered listeners and provoked suggestion of violence;
Energetic, even raucous, protesters who annoy or anger audiences'
Demonstrations that slow traffic or inconvenience pedestrians;
Peaceable protest activities, even when some in the demonstration might have transgressed the law;
Actions that have been held not to be protected thus justifying police intervention, include:
Violating a regulation of the time, place, or manner of speech that is narrowly tailored to serve the government's legitimate, content-neutral interests;
Demonstrations or protests where "clear and present danger of riot, disorder, interference with traffic upon the public streets, or other immediate threat to public safety, peace, or order, appears."
Where a public gathering threatened to escalate into racial violence and members of a hostile crowd began voicing physical threats;
Defying appropriate police orders to cease and desist;
Makin threats of physical harm to police or members of the public;
Inciting violence or disorder with a direct appeal to imminent lawless action;
Imminent disorder when coupled with petitioner's deliberate defiance of the police and their orders to disperse;
Summary
While there is no single bright line rule on what constitutes "peaceable assembly", the test for the "clear and present danger" of violence or lawless action seems to come closest. In mot cases police must unless there is an immediate prospect of harm, order an assembly to disperse and give them a reasonable chance to do so before making arrests, particularly forceful arrests. Failure to do so may subject LEOs to personal civil liability.
