Normally the bike will start jumping out of 2nd and quickly end up with no 2nd gear at all. Here is my method that I use to change the selector mechanisms on FZS600s....
ok if you have a Haynes manual/book of lies ignore it completely from here on in:
Drain oil.
remove down pipes
remove sump
remove clutch casing
remove entire clutch basket (withdraw push rod and don't lose the ball bearing that sits in the basket end)
remove oil pump
remove the gear lever
remove gear selector shaft
remove the sprung wheel that the selector drum runs on (making sure that its in neutral and taking note of how the mechanism goes back in)
remove the bolts that hold the fork shafts in place
Now the next section is easier if you have a bike bench otherwise you have to do it laying on your back
slowly withdraw the rear rod that the left and right forks run on (the forks should not come out but just make sure you steady them as the rod comes out so the don't wedge against the gears)
repeat with the front rod that holds the centre fork
now carefully remove the drum and remove the forks as the drum comes out.
make note of how the forks go in (they are marked left, right, and centre)
now clean and examine the drum and forks. normally the drum is fine but the forks especially the centre will have been worn and it will be obvious which is causing the problem, REPLACE ALL THREE, the drum can normally be reused.
Now replace the drum but don't secure it in place just yet but make sure neutral is pointing down to the six o'clock position
locate the two new rear forks then slide the rod into place guiding the forks into the correct position as you go
do the same to the front/centre fork.
refit everything inside the engine making sure that the oil pump shaft mates securely with the water pump shaft when refitted
cycle through the gears by hand before refitting the clutch casing just to make sure
refit clutch casing, sump, and pipes, fill with oil
strut around the garage in a manly way and sup tea whilst bragging to your mates/wife/kids that you are a truly great spanner monkey!
Job done!
**NOTE** This is not a novice job and I may have omitted and item or two as I am typing it from memory but that will become obvious if you attempt the task. My local Yamaha dealer still insist its an engine out and split job but I have done this for several of my customers without issue, but saying that unless I am actually doing the job I do not take responsibility for anyone using this post to fix their own bike, so if you arse it up don't blame me..... Thank you.