When I accelerate hard, after I clutchless shift from 1st into 2nd gear, I'll get kicked back into neutral, after getting into second gear briefly.  Is this indicative of sloppy shifting or is it symptomatic of a mechanical problem?  I'll be accelerating in second gear for about a second, then it gets kicked back into neutral.

I won't hide my bias on this issue.

It might be symptomatic of bending or grinding down the dogs, or the holes they fit into, due to clutchless shifting.

It seems that a certain segment of hardcore riders recommend clutchless shifting, and nearly all mechanics recommend using the clutch every time you shift.

I trust the mechanics on this one.

If you apply power while the dogs are partially engaged (as it might do if you're not using the clutch or if you shift weakly), it can slip out of gear (and coincidentally, do more damage).

Practice using the clutch until you get better at it.  You should be able to use it every time you shift without severly impacting your quarter mile time.

It's harder to get a clean and smooth clutchless shift to 2 than it is to any other gear.  I don't do it unless I'm booking it.  I would say you're not executing the shift properly.

Sloppy shifting, unless it starts to happen all the time then its damage to the shift dogs caused by sloppy shifting...

it starts as sloppy and ill timed shifts and throttle work, then works its way to bent shift forks and rounded Gear Dogs.  

try this, start of normaly, get a few revs and use the clutch and shift to second, once in second roll with light throttle to about 4k rpm then lay it wide open to about 10k, do this a couple times. if it doesnt come out of gear/ false neutral, then its LLT syndrome,   if it skips then grabs, once or twice, then its time to talk to me about buying one of my spare trannys in a box.

I didn't know what LLT was...

http://www.gstwins.com/forum/viewtopic.php?t=21435

I've got a new GS500F with only 2800 miles, I use my clutch when shiftting and I get kicked out of second gear too. It seems to happen when I keep R.P.M.'s low ( like when your in stop and go traffic jams) and its dropped out a couple of time when I've opened her up. Its never happened when I shift under normal driving or if I get on the throttle just a little hard !  Its not my first new bike, have plenty of experiance riding and maintance. I did use full synthetic oil on first two oil changes, does anybody think my gears just need to wear in more or what. Any ideas ?

Sounds like lazy left toe to me..... use more toes

Here's my wife's explanation (a non-rider):

"Maybe you're just a Bad Person?"

(Hey, just passing on one of a vast number of possible explanations...)