After teaching three lessons in a row today and spending a fair amount of time on scales, it's hard to avoid thoughts of the ugly beasts. The scales, of course. I've done them and so have you, and we both know someone who has done more than we have. How much is enough, or too much?
On my site natesviolin.com I tell the story of my first lesson with Felix Galimir, where I learned that I would be spending a couple hours on scales every day until further notice. Like most tasks that seem (or are) monumental, this yielded lots of good fruit. I didn't exactly keep it up, but a normal practice day has me doing between 30 and 45 minutes of scales.
For my students that don't have pressing repertoire deadlines, I don't mind if they devote a third or even half their time to scales, as long as they are showing good improvement and increased fluency with them. For those who have to get through more repertoire, I think half an hour is a good minimum. This includes arpeggios and double-stop exercises, of course.
After working through some of Ruggiero Ricci's left-hand book and noting his love of thirds, I've found a big daily dose of thirds very helpful for the hand and the ear. It's impossible to have a bad hand frame while you're negotiating passages in thirds, so this can really iron out some of those issues.
I don't teach a scale system, although I use the Flesch book because of its completeness. In my own work, I play single-note scales in separate bows, both on and off the string, and slurred (free bowing). Arpeggios, same thing. Thirds, then fingered octaves (static 1-3, static 2-4 and finally 1-3 2-4).
For the single-note scales I like to work through 6 to 12 keys, and for the double-stops just two or three. I leave a key and then come back to it a few minutes later to see what sticks. The idea is to keep things moving and to constantly vary the game. 60 or even 30 seconds of focused work on a scale will be very productive, but don't be surprised if your mind wanders at that point. A new key or a different kind of scale is essential.
And if your arm/hand gets fatigued, take a break! No good will come of continuing to pound away at something past its time. That's how I've hurt myself the few times it's happened. Thank you very much, Paganini 24th caprice! Actually, it may have been Bach or something else. It just sounds more valiant if it was a caprice.