Two points.
First, wall transformers are not regulated, so they may put out too much voltage and damage the lamp. There is no danger of burning down your house. Get a big enough one, connect up the light, but check the voltage on the leads when it's running. (You could get the type that are sold for cars, they should be designed for more voltage fluctuation, but may need more power and cost more.)
Second, PC power supplies check the load and won't work if there's not sufficient current draw. I used to use an old dead hard drive to test power supplies. I don't know how much power the cold-cathode light draws; you might need to connect up several of them. Edit: Get a used AT-type supply, not an ATX one - the kind with the Big Red Switch on the side.
What I'd suggest is watch for a hamfest in your area, then go there and get a used open-frame 12V regulated supply. You should be able to get it for about a quarter. Hook that up and run all the lights you want.
BTW, why do you want to put cold-cathode lights on your desk?