I gotta say, I've got Klusons on my Jazzmaster and I the trem quite a lot. No issues with it losing tuning while I've played it, besides the occasional G-string nudge, which has been necessary on every guitar I've ever played.
I used a roller tree, with the bottom ground down to a reasonable height (stock, those things are too damned tall to give you the proper break angle beyond the nut), put graphite in the nut slots and have a Mastery bridge.
Aside from some minor grounding issue caused by my novice wiring skills, the guitar is a dream. It plays beautifully, light trem use doesn't throw it out of tune. I'm pretty sure I paid $50 for a Fender trem, but Cagey's right that you can put that $25 toward the quite expensive -- but well-worth-it -- Mastery. What a beautiful bit of machinery, that thing is.
Edit: Oh, I use .011 - .050s on mine ... and I've pondered going up to .012 - .052s with a wound G. I like heavier gauge strings, in general, but I find that they really suit the Jazzmaster, in particular.