If you're in the US and shooting 24p then 1/48 is the norm. Basically 1/48 is equivalent to a 180 degree shutter angle which is typical when shooting film. 1/60 would be for when shooting 30p and 50 for 25p (PAL users). If you were shooting computer screens or under fluorescent lights then you'd use 1/50 or 1/60 depending on your power Hz rating to remove flicker.
The often given example for different shutter speeds is saving private ryan and the effect of seeing debris flying when there are explosions. This is the result of a faster shutter speed bringing fast moving objects into sharp focus. Slower shutter speeds will introduce blur.