Used Bike
Recently (well, a little over two years ago) I bought a used motorcycle. It was simple enough. After finding a good example of the model I wanted in an internet search, I phoned the dealer who was selling it, and arranged to inspect it three days later. Then I went to the bank and got a bank cheque printed. I flew to Tauranga on a cold day in Winter and had a perfunctory look at the bike, then paid for it and bought a new helmet and overtrousers and gloves. Then I got on to it, checked that it had fuel (it was full) and rode it home. Four hundred and forty kilometres in Winter on a model I'd last ridden twenty-four years earlier on a racetrack. I got away with this because it's a BMW.
I'd returned to motorcycling less than two years before that, after an eighteen-year break, with a 1989 Jawa 350 which performed better than I'd expected. At the time I also had a car, but when I bought a house at the end of 2005 I lent the car to a friend who had helped me to move my furniture into the house, as he was having a few car problems. He needed it for more than a year, and when he returned it the poor thing was terminally ill with salt poisoning (He lives very close to the beach). Having found two wheels adequate for my needs, I decided I needed a bigger bike to cope more effectively with the strong winds characteristic of this part of the coast. I've always liked BMWs, so much so that in 1976 I'd bought a Ural 650 because I couldn't find any used BMWs and I couldn't afford one anyway. That was a mistake. It never left me stranded a long way from home, but that was only because it wasn't reliable enough to carry me a long way from home. Lesson 1 for used bike buyers: Know what you want, and avoid substitutes.
The next thing to consider is the kind of use you intend to make of it. If you're going to commute on it and occasionally go to the next town (and I'm assuming you're not in Saskatchewan or Western Australia, so the next town is only, say, 20 or 30 kilometres away), you won't be wanting a Ducati 996 or a Suzuki Hayabusa. If your beloved lives hundreds of kilometres away, you should be looking for comfort, reliability and good range. And a fairly large engine with abundant torque at medium speed. You get the idea.
Mileage is mostly irrelevant, provided that all mileage-related work has been done when it should have been. Condition matters more. Find out as much as you can about the bike you're looking at. If it has a reputation for easy wheelstands, check the steering head bearings and fork seals. If a bike spends too much time with the front wheel up in the air, it's likely that the oil will fall away from the pump pickup. This can cause expensive trouble. Round up the usual suspects, Chains are expensive and a new chain should, ideally, be accompanied by new sprockets. No, I don't go to that expense either, unless it's definitely necessary. Worn swingarm bushes will kill a chain faster than anything other than worn wheel bearings. Check both. Look at the chain too, and if it's dry, walk away. Look for oil leaks. Check the fuses. If they are heavier than specified, ask why.
If you're looking at a fairly old bike, like a 1980s BMW, look around to see how many are still around. If a once-common bike, say a CX500 or a GS550, is hard to find now, check the availability of parts. It's generally easier to find and pay for parts for a Norton Commando or a BMW R75/5 or a Triumph Bonneville than anything Japanese from the 70s or 80s.
So, when looking at a used bike, you need to assess condition, suitability for your intended usage, parts availability and price, and model idiosyncrasies. Enjoy the search, and if in doubt, consult an expert. There are plenty around.
And finally, a couple of tips. Airhead BMWs expect new gearbox bearings every 100,000 kilometres. If you can hear them, they need replacing. And if you want a used Ducati and you have good engineering skills (e.g. a toolmaker, like me), buy one that's due for its 20,000 kilometre service, using that fact to lower the price.