Why does your 1995 Yamaha yzf 600 keep jumping out of second gear?

mine doesn't do that... have 30k miles so far, if your bike does that seems like the fork that shift between gears have worn out. the 2nd gear on any bike always take the biggest abuse among all other gears, because when you shift from 1st to 2nd you have to travel the gap from neutral that give the 2nd gear harder jobs. overhaul the transmission and replace worn out parts will fix the problem, but they are pricey unless you can do it yourself. best deal would be bear with it, jump from 1st all the way to 3rd. YZF have close ratio so it should not give you big issue unless you ride in the city a lot... or you can try to hold the shifter right after you shift from 1st to 2nd, not suggested but work with my friend's FZR...

Good Luck