Disclaimer: there are whole sections of criminal law courses that deal with this issue and the various laws and precedents that have shaped american jurisprudence relating to right of privacy, expectation of privacy, and search and seizure. So, this is more of a primer than a comprehensive answer.
That being said, the Fourth Amendment, which protects the people from unreasonable search and seizure by the government, dictates that there must be probable cause for a search warrant to issue. However, in addition to the restrictions that are imposed by the Fourth Amendment, several states’ constitutions, as well as a variety of state and federal statutes, rules of procedure, and law court rulings interpret these edicts further, limiting the process for obtaining a search warrant (a state can add additional safeguards, or impose more stringent requirements, but not less than the 4th amendment demands). On the federal level alone, the laws on search and seizure are also limited in scope by sections appearing in Title 18, Part II, Chapter 205 – Searches and Seizures. 18 U.S.C. § 3101-18. Federal warrants are further governed by Rule 41 of the Federal Rules of Criminal Procedure.
Only Judges and magistrates may issue search warrants. To obtain a warrant, law enforcement officers must show that there is probable cause (i.e. grounds) that search is legally justified. Officers must support this showing with sworn statements (affidavits) under oath, and must describe in particularity the place they will search and the items they will seize. Judges must consider the totality of the circumstances when deciding whether or not to issue the warrant. When issuing a search warrant, the judge may restrict how and when the police conduct the search.
Police officers (a broad term I'm using for ease of discussion to include all manner of state and federal investigators operating under color of law) seeking a warrant do not need to show that the people being searched (or those whose property is being searched) actually, or even probably, committed the crime(s) in question. Rather, officers merely need to show probable cause exists to show that the evidence sought-after is likely there and upon discovery may implicate some person or persons in criminal activity (that has been committed).
A landmark case that resulted in some reforms is Zurcher v. Stanford Daily, 436 U.S. 547 (1978), in which the Supreme Court opined that a search conducted by the police at a student newspaper, where the newspaper was not implicated or connected to any criminal activity, was a legal search under the Fourth Amendment, because the police suspected it had photographic evidence of the identities of demonstrators who assaulted police officers. Afterward, however, some jurisdictions responded by passing laws restricting or forbidding these kinds of searches - so again, what stands as acceptable procedure is not static.
While you may not often hear of requests for warrants being denied, it does happen if there is insufficient PC pled in the affidavit; General unawareness is not surprising as it is not something often made public, although there are instances. Most officers know when they have enough for a warrant and know how to adequately swear out an affidavit so that PC is present enough for the judge to sign the warrant in good faith. However, some Judges are more stringent and demanding than others with regard to how the PC was obtained,the scope of the warrant under those facts, etc. Certainly, not every application is granted.
Cornell has a great primer of this issue and you can find article when application for a warrant was denied...especially in higher profile investigations.