You don't even need to build up a new wheel or buy only a bike with horizontal dropouts like the other posters suggested. With a vertical drop out bicycle, just buy a gusset singlespeed conversion kit and a Surly singulator chain tensioner and blam-o instant single speed conversion. No need to redish the rear wheel. There's cheaper versions of the singulator made by other companies too and you can even go to a shop and ask to see if they have old cog spacers to fill up the cassette instead of buying the Gusset conversion kit. I did this on a modern road bike with vertical drop outs and it works great.
If you live in a relatively flat area, singlespeed is ideal for the winter since it is so low maintenance. Just clean the chain and lube it up/wipe off and you are done.