I carve my bridges to show about 3 mm under the halfway stop (where the 12th fret would be on a guitar). This is probably a higher action than some would prefer, but it suits me. For [email protected] : "action" is the clearance of the strings above the frets of a guitar; I cannot recall the equivalent violin term.
The following document may be of interest to you:
Violins do not suffer fret buzz like a guitar can, but insufficient string clearance can adversely affect intonation of fingered stops, and too much will make playing uncomfortable and bowing hyper-sensitive to angle.
"It is vain to do with more that which can be done with less" - William of Ockham
"A crown is merely a hat that lets the rain in" - Frederick the Great
If the action is too low, the G string in particular may buzz when played loudly on a stopped note -- the vibration may be large enough to hit the fingerboard. But I like a relatively low action anyway.
String height is typically measured at the end of the fingerboard. On my viola, I have my A string 3.5 mm above the fingerboard at the end, and my C string 5.5 above the fingerboard at the end. That's half a millimeter lower than what is typically recommended for a viola, but not so low as to cause intonation problems and buzzing lower strings. The lower action is especially helpful for me because I have very short fingers.
(A typically recommended violin clearance at the end of the fingerboard is 3.5 mm for the E string and 5.5 mm for the G string. Most factory violins arrive with the bridge cut a bit too high, probably because you can cut the bridge down but can't raise it without getting a new bridge.)
AndrewH pretty much nailed the answer I would give. To add some confusion, perlon strings need more room than steel strings and light tension strings of all types need more room than medium tension. Heavy tension need the least amount of room.
Fast players therefore often use steel heavy tension strings.
Success is the progressive realisation of a worthy ideal. —Earl Nightingale.