I get nervous when I take the ice and assume the starting pose, so I actually practice that, to be sure I can hold it still for a long time.
Once I'm skating, the nervousness goes away and I concentrate on the program.
I first warm up off-ice and then on-ice. It doesn't bother me to watch others skate, even my competitors. Walking around, little hops, leg lifts, waist twists, etc. keep me warm.
When I skated ISI, I used make a mental note of which elements the other skaters did better/worse than my usual performance. During my skate, I'd try to make those elements perfect.
If I had to wait more than a few minutes, I'd close my eyes and walk through my program in my mind. When Sony released the Walkman, it was wonderful - didn't have to put my fingers in my ears to block out the arena music, lol.
Although my family wouldn't agree, I'm very calm and determined rink-side as a coach. I have them warmup off-ice, then take the ice for the event warmup, which we've planned the week before.
While they wait their turn they walk, stretch, balance and do gentle calestenics. (Waist twists, toe touches, leg lifts, knee bends)
What's more important is going into the competition or test session prepared. Extra run-throughs, mastered skills rather than stretch skills, consistent practices and good planning really help exterminate butterflies. I have my skaters practice their moves in every possible setup, so that when the judge asks them to use the red line with the logo rather than the blue line, they don't get rattled. My being calm, rink-side, also helps because skaters pick up on nerves. YMMV with stuff like stress balls or worry stones. One of my skaters insists that I have to hold her guards while she skates. If I put them down, she insists that she'll fall, lol. I humor her and hold ALL my skaters guards while they perform. It's harmless.
I want my skaters to be able to enjoy their 1:30 in the spotlight, so keeping them calm and composed really helps.
As a side note: I don't emphasize where they place or what medal they receive. I'm focused on their performance - clean elements, good transitions, smooth skating and dramatics.