I've restored a few bikes. Rather than brands per se, I look for a particular range of frame and component standards that let me mix and match available parts without either going broke, or going crazy trying to find some weird thing. I've found that steel frames from the 70s through the 90s with horizontal dropouts are kind of a blank slate, especially if they have the clearance for wider tires that are preferred these days. With steel, I can play around with things like dropout spacing for different components such as old gearhubs.
Components, likewise. Square taper BB's and cranks give the widest selection of second hand parts. Of course a bike that already has those parts attached is more valuable, even if they end up on another bike or in the bin for another day.
After a near-miss, I will probably avoid bikes that don't come with a seatpost, and I'll make sure the seatpost isn't stuck. I got a nice Schwinn frame sans seatpost, and it was only a stroke of luck that I was able to get the right measurement and find a second hand part that fit just right.
Just looking at the ads on the Craig, it looks like bikes that are sized for the average size male, and smaller, are the hardest to find. There are lots of people selling super-tall bikes that I just have no hope of riding.