For showmanship, trotting circles, trotting through serpentine cones. Short distances to trot.
For equitation, two-track, hand gallop, no irons.
For horsemanship, Here's a link to a pattern I did years ago that didn't look that hard, but had numerous sticky points in it. The pattern: lope left lead to two. Stop. 2 turns on the haunches, second one faster. Extended trot with two square corners. Extended lope at 3, circle right. Collect to regular lope to 4. Stop and back.
I've got a huge book of patterns I've collected from shows. If I think of any others I'll let you know.