Comets aren’t exactly a dime a dozen; it’s a great matter of pride when someone finds one. But there are dozens discovered every year. I believe that I’d read once that there’s a society of comet-hunters, though I don’t know what it’s called.
Pantellerite: please note bibliophage’s comment – you’ll need a camera attached to that telescope. You’ll be looking for fuzzy objects in the night sky, the blurring indicating rapid movement. In most cases the film is cooled by dry ice to maximize collection of the faint light.
There’s an excellent interview with Carolyn Shoemaker here that explains her strategy for seeking comets and a number of other items (telling a near-earth asteroid from a comet):
You can find several excellent on-line tributes to Eugene Shoemaker with a simple Google search. You may also be aware that the NEAR asteroid spacecraft was named NEAR Shoemaker – the one that they recently landed gently on the asteroid.
I had wondered if Carolyn & Eugene had found any more comets before his death in a car accident in 1997 on a remote Australian road. Apparently they had not, though Shoemaker-Levy 9 was spectacular.
For his part, David Levy has at least 21 comets.
It’s a fascinating area. If I didn’t live in Seattle I’d probably put in a backyard observatory!