I didn't have time to read through all the other replies, but have a few mins to write what I can remember.
Note that the K2 and K3 models had a fixed screen. It's difficult to sort out any buffeting on these. From K4 onwards they had the adjustable screen, same as the old DL650 (3 height positions and an undocumented 4th position.) It's worth mentioning that people shouldn't be put off by the buffeting, a Madstad bracket (for K4 onwards) or another screen can sort this out quick.
Other common issues is the clutch chudder, there's a bloke on this forum who reworks the clutch baskets to fix the issues permanently.
Emission regulations got in the way and Suzuki remapped them to obey, between ~2500 and ~3500rpm they can run a little rough (lean.) 5 minutes with a Yoshi box and this can be tuned out so it's not an issue (IIRC it needs +5% fuel in low and medium revs, or was that just low?)
At higher speeds (over 100mph) the Vee can get a bit light in the front. Dropping the forks or jacking up the rear a little changes the angle that the headlight cowl cuts into the air and alleviates this. It's not the right bike if you want to go that fast anyway.
The Kawasaki KLV1000 is a K4+ Suzuki DL1000 with orange panels and different stickers, otherwise it's practically identical.
Stock exhausts have been known to catch fire so getting one with aftermarket cans fitted not only saves a lot of weight, it safer
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That said, it's not a common thing, just occasional, but they are heavy as a pair of bomb shells.
Water pump cover corrosion and clutch slave cylinder seals leaking have probably been mentioned. Both are easy to fix. The rear throttle body rubber boot can sometimes come undone, it will suck in air and pop and fart, pretty easy to fix if you take the panels and tank off. Best to strap them down with some wire/cable ties to avoid it returning.
It's worth mentioning that there is much more right about the Vee than there is wrong with it. Lights are excellent, they are very stable on the road and the ride is comfortable. Fitting a fork brace tightens the steering up and they are a hoot because of all the stomp on tap. You have a very commanding riding position so you can often see over the hedges on little country roads to spot oncoming traffic and the flow of the road where an arse up sportsbike rider wouldn't have a clue what's coming. Pillion accommodation is a favourite among most wives, girlfriends and mistresses. You can usually pick one up for cheap, spend a couple/few hundred quid on them and have a great everyday bike as well as a good tourer. Parts are easy to get and they are still an easy bike to work on. Advice is even easier to get, right here on this forum
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