Would a permanent resident be asked to do jury duty, or must a juror be a citizen? What are the rules of who is eligible for jury duty? Are there people in the US who go their whole lives and never do jury duty or is it usually the case that a person does it at least once in his/her lifetime?
For federal trials, 28 USC 1865(b) states that the judge
shall deem any person qualified to serve on grand and petit juries in the district court unless he (1) is not a citizen of the United States eighteen years old who has resided for a period of one year within the judicial district
There are similar requirements in other states, for example in Washington, RCW 2.36.070 which includes being a citizen as a mandatory qualification. There are typically also age and residency requirement, requirements for competency in English, mental capacity also felon-exclusion provisions and charges-pending exclusions. The qualifications are jurisdiction-specific.
Many people never serve on a jury, indeed many are never called to serve on a jury. There is no reliable count, but polling indicated that around 1/3 of adults have served on a jury.
You are enrolled for the pool of jurors upon registration to Vote, which is a benefit to Citizenship in the United States, thus permanent legal residents are not eligible to serve on a jury.
In the United States, when a Jury is needed, the court will form a pool of potential jurors from the population of eligible jurors in the jurisdiction of the court (A jury pool for simplicity sake). This pool is a large group (the size varies depending on the nature of the case and the controversy of it. Generally big media circus causing cases will have a much larger pool as the number of people who likely heard of the event and thus may be biased is significantly increased.). From this pool a jury of 12 Jurors and a number of Alternative Jurors (typically at least enough to cover a significant number of Jurors, if not more.) are empaneled and the trial commences. Both the Jury and Alt Jurors will hear the full trial, but the 12 Jurors will deliberate on the verdict. The alternative jurors will only deliberate if a juror is dismissed for whatever reason.
Whether you will serve on a jury or not largely varies from individual, the population of the jurisdiction, and other factors. For example, in one stat I lived in for only four years of my adult life, I was summoned once. In the state I lived in for the bulk of my adult life, I have yet to be summoned. Even then, that doesn't mean I will be empaneled. Disqualifying someone from Jury duty is very easy as both attorneys can motion to dismiss for a number of reasons and they are given several "free" motions to dismiss for reasons of "because I don't like this person as a juror".
Some reasons to dismiss a juror, but not all of them, include familiarity with the case, familiarity with similar cases, previous relationships with any party to the case on either side, previous relationships with another empaneled juror, certain occupations, namely those that are related to law enforcement, lawyers, judges, court clerks who would have a much better than average understanding of the law, familial relations to people who work in those industries, work place obligations that would be burdened if you were sequestered (they don't want you to give a verdict that will get your jury duty over with because if deliberations run too long you will be fired), home situations that are unduly burdening your family (single parent raising 5 children, the oldest of which is in elementary school), personal life situations that would unduly burden you, (your going on vacation three days after the trial is expected to start), certain types of entertainment (similar to work in legal or law enforcement, but instead of knowing more than the average person you THINK you know more than the average person), certain shows (during the height of its popularity, fans of CSI and spinoffs were disqualified because they would often demand more evidence than was necessary to convict), biased (in the modern courts, if you are distrustful of people of a particular race/religion/ethnicity/culture/ect that will be expected to have representation from the trial party, they don't want your bigotry biasing the decision), certain political beliefs (you are disqualified from death penalty cases if you are in favor of abolishing the death penalty because your likely to give a not guilty verdict to spar the guy from the chair, despite evidence saying he is guilty as sin. In some jurisdictions, the jury must find that the defendant is both guilty of a crime eligible for the death penalty AND then must determine if they want to recommend the death penalty. If the case involves crimes that you do not think should be crimes, you won't serve), having been convicted of a crime, having been a victim of a crime, having a close relationship to someone who was convicted of a crime or a victim of a crime, knowing of certain things a jury can legally do (like Jury Nullification... especially Jury Nullification), wanting to be on a jury, failure to appear for a past jury summons (this is a crime, as a matter of fact), lying on your questionnaire to determine if you should be allowed on the Jury (this is a crime, perjury to be exact), and lying on your questionnair to get out of Jury Duty (if you do it correctly, congrats, you got out of jury duty, if your caught, congrats, you've shown you clearly won't give the trial the serious time it deserves. Also perjury.).
It may seem like you have some contradictions (such as being a criminal or being a police officer) but again, it's to eliminate bias so that the defendant gets the best impartial trial possible. And some conditions are something of a permanent disqualification... like being a victim of a crime.
As has been observed by cynics, In the U.S. nobody is convicted by a jury of their peers* but a jury of 12 people too stupid to get out of jury duty.
Edit: In addition, in the U.S. a vast majority of legal cases are settled out of court and never see trial, and thus a jury panel, and under certain circumstances a jury can be forgone and a judge can do the juries job for them. All that in mind, it's entirely possible to live one's entire life and never serve on a jury. Or you can serve on more than one. In the OJ Simpson murder trial, there was at least one juror who the prosecution attempted to disqualify after being empaneled because she had sat on several juries and was noted for arguing successfully for convictions in all of them.
*Note: The U.S. constitution does not say "Jury of one's peers". That line is from the Magna Carta, and refers to peers in the sense of Peerage. The U.S. doesn't have peerage... though in Court Martials, the rank of the jury panel matters in relation to your own.