What shutter speed would you recommend for an indoor karate tournament with slightly dim fluorescent lighting? I will not be using a flash. I'll be using a Canon T3i with an 85mm f/1.8 lens and using shutter priority mode. I normally use raw format but I'm switching my camera to jpeg format to avoid processing lag between pics. What do you think is the slowest shutter speed I can use without motion blur?
"...without motion blur" is somewhat subjective, as the acceptable amount of blur for me might be different for you. It also depends on how far your subject is from you. Because of these two factors, a specific shutter speed is not what I would recommend looking for. Instead, and since you are shooting digital - simply experiment and see what shutter speed works best for your needs.
Very generally speaking, I would start at 1/125s second shutter speed and work faster from there. You could potentially need as fast as 1/500-1/1000s to freeze very fast motion. I would say a safe bet would be 1/250s, but your results will vary. Depending on the routine, some movements could be very fast and you might want every bit of the subject sharp, if that is the case you will be testing the limits of your equipment with certainty. If the routine is a bit slower or you find a bit of blur in the extremities acceptable, this should be quite achievable with your kit.
What you may find, is that you are forced to shoot at 1/125s because that also requires either a very narrow depth of field, or a very high ISO that is too noisy for your tastes. Again, personal preferences play into this.
Keep in mind also that some motion blur can be an artistic effect that adds to the image. Especially if you can shoot with a flash(but maybe you aren't allowed to), and use second-curtain sync.
Finally, why not browse through some Flickr images and examine the shutter speed used? Nothing beats real world examples to show the differences: http://www.flickr.com/search/?q=karate
Here are some examples at varying shutter speeds. These are of a soccer game, but the same idea applies. These are all pretty poor images unfortunately.
18mm, f/3.5, 1/40sec: 50mm, f/2.8, 1/125sec: 18mm, f/3.5, 1/500sec:
Presumably you want to freeze motion, keep your entire subject in focus and minimise noise - in that order. If you want less noise, you'd have to accept some blur (which as dpollitt said, can turn out nicely) - the blur could be due to motion (slower shutter speed) or focus (narrow depth of field/large aperture).
To start, I'd set a shutter speed that will freeze the action, so I'd probably go for about 1/250 as a minimum. From there, I'd set an aperture that will keep my entire subject in focus - if you're not sure what aperture to choose, try using dofmaster.com, or simply experiment on the day. Finally, I'd raise the ISO to whatever was needed to ensure the subject is appropriately exposed.
If the lighting stays more or less the same and the general composition of your photos don't change, you should be able to stick to the same settings most of the time. However, beware that if you're not in full manual mode, changing compositions and having different background colours can force your camera to over- or under-expose your subject.
Two basic rules about depth of field: 1) The closer you are to your subject, the narrower the depth of field, and vice versa, for any given aperture; and 2) the larger the aperture (smaller f-number) the narrower the depth of field, and vice versa (larger f-number), for any fixed distance to your subject. If you decrease the aperture to get more into focus, you may need to increase the ISO to unacceptable levels (heaps of noise), but that depends on your taste.
Having said all that, please experiment with different settings! It's the best way to discover what does what and could provide you with some creative insight for future shoots.
EDIT: In response to Robin's comment, that's correct. To capture a subject about 1.8 meters tall with an 85mm lens on a crop frame sensor, you'd need to stand about 10 meters away, giving you about a meter depth of field at f/1.8 But keep in mind the camera focuses at the surface of your subject, which has depth itself. A person isn't paper thin. f/1.8 at 10m away from your subject, on your lens, would give you only half a meter of acceptable focus into the subject. With extended arms and legs, you'll want to have an even deeper depth of field. An aperture of 2.8 should do the trick, and if you get closer to capture only half-body or head portraits (which often sport photographers do) you'll probably want to decrease the aperture a little further.
This is all very theoretical and probably not necessary - I'd say fiddle around with the settings until you're getting your desired results.