Especially in supported pirouettes, I've seen a large variety of arm positions, some less beautiful than others. (I mean ordinary pirouettes when the choreography doesn't specifically have the arms in fifth or anywhere else.) Do different schools have different ideas of where to put one's arms during pirouettes (of any kind)?

Yes.

Drawing on memories I haven't used for a long time

I was taught (at the Joffrey) that they should be in first position for your standard pirouette.
you plie in fourth, with one arm curved in front and the other out in second (term? fourth en avant?), then plie on both legs and as you reach passe the arms come together.

When i went to SAB, i learned to cross your arms tight to your chest in turns. The preparation is also different...WHile you start in 4th the back leg remains straight and the arms, which are curved in most techniques, are stretched out, as if you are in arabesque. Then brought in tight as you reach passe.

Sorry if this is unclear.

If a teacher tells you to pirouette with the working leg in retiré position without saying/demonstrating a position for the arms, it is usually safe to assume s/he expects 1st position. For pirouettes with the working leg in an open position (aka grands pirouettes) the arms would usually be held in Vaganova 3rd position if the leg is extended to the front or back and in 2nd or 3rd if the leg is held to the side. For a pirouette en dedans with the working leg in attitude derrière, often the arm that corresponds to the working leg is raised to 3rd position while the other is in 2nd. There are, of course, no hard and fast rules, but these arm positions are very commonly used for these types of pirouettes.

Thank you both for clarifying.

I was partly moved to ask this question by the extraordinarily cavalier way in which Margot Fonteyn flings her arms out in preparation for supported pirouettes in the film of Sleeping Beauty Act III.

Hans, is Vaganova 3rd sometimes called 4th position in other training? I'm slightly confused.....

For supported pirouettes are the arms usually wrapped, then? I assume a proper 1st position is impossible without inflicting pain on both parties involved.....

Vaganova 3rd arms is usually 5th in other methods.

And wrapped is just too far in for most supported pirouettes. First is correct, but perhaps only a little pulled-in to avoid the partner. After all, you're only usually doing a triple, tops.