ASMs have a large role in my school! Here's what my ASMs will do in our upcoming production:
During Rehearsal Time:
-Block at every rehearsal, and review blocking notes at the end of each week to make sure everybody is on track; discrepancies are to be reported to me, the SM
-I have three ASMs, so I have one take notes for props, one for costumes, and one for tech and paint. Our director likes to shout out random orders during rehearsals and expect them to get done, so I can't do it all myself; ASMs then give their notes to me, which I compile with my own, which I then bring to the director and crews to discuss.
-Stand in or read lines for absent actors without understudies
-Answer blocking and lines questions for actors if I am otherwise occupied
-Arrive fifteen minutes early to every rehearsal to unlock the theater, sweep, preset, check rehearsal props, and other miscellaneous tasks
-Assist with crew work (tech, lighting, sound, paint, etc.) when necessary
During the run of the show, I have one ASM in the lighting/sound booth with myself, the AD, and the director; one SL; and one SR. The ASMs in the wings are in charge of prop table maintenance, or if there is a prop runner or the prop mistress present, simply be on book. The ASM in the booth is to be on book, and if worse comes to worse, can walk around backstage to solve any problems that may arise
In summation, my ASMs are very much SMs in training in that they do what I do, only with the work split between the three of them. I try to teach them whatever and whenever I can, and usually they learn a lot (=
Number one SM-ASM relationship pitfall: having your ASMs buy you, your actors, or your director food. AVOID at all costs, for it breeds resentment.