I use either a Sunnto Tandem or Suunto Twin (similar to the tandem). I do what Scott and Jeff do and just tuck it in my suit/clothing, depending on what I'm wearing in warmer or colder climates.
I've seen others using the neoprene cover idea, and it's a nice way to do it. I hadn't heard of the Survey Bra before, but it's nice to learn something new every day here on the forum.
I store my instruments in the pouches they come with, but I make a little hole in the upper corner of the pouch and thread the lanyard through it. That way the pouch stays captured on the lanyard when I'm surveying. I also keep a green keyring style LED light clipped on the lanyard, which I use for lighting the instrument and for marking stations.
If I'm carrying instruments but not surveying, they're usually in my pack. If you think juggling a compass/clino is tough moving through a cave, try using one of those larger Disto LDMs as well!