How do you go about mic'ing a guitar amp for live performance? Would you do anything different if you were mic'ing the amp for studio recording? Is there anything specific I should look for in a microphone for either of these scenarios?
Honesty? Any way you want, any way that sounds good to you.
That said, there are a couple of tried-and-true techniques. For live performance, you can't do much better than a simple Shure SM-57 in front of your best-sounding speaker, maybe off-axis if you prefer that sound. 57's sound great and handle the bumps and bruises of live gigs exceptionally well.
Personally, I would never mic an amp. It doesn't make any sense to me when you could just run a line from the amp (or straight from the guitar/box) into your recording equipment and have higher audio quality.
You might want to ask this question on Audio.SE as well, they may provide better answers.
There are lot of ways; if you have and amp/cabinet with more than one speaker cone; mic up the best sounding one; one or two dynamic mics will sound very different in different areas of the cone so mess around with it; also; when dialling in your tone on the amp; get your ear as close the the speaker as possible; so you can hear what the mic(s) hear (its very different from the the sound you would hear standing away from the amp so you'll have to adjust accordingly to get the tone you what recorded). Some people also stick a mic around the back of the amp.
Besides miking to accurately reproduce the sound of your amp/speakers, you're also trying to isolate your guitar sound from the rest of the band, so mike placement has to take that into account.
Don't point a mike toward another instrument's amp or toward the drummer, because bleed-through from the other sound source can mess up your sound capture, and make it harder on whoever is running your mix. So, if you put a mike off-axis from the speaker, keep it reasonable.
If you have a good on-stage monitor system, you might want to consider putting your cabinet off-stage with a mike (or mikes) so you can crank the amp head without blowing your ears out. Having it offstage isolates the sound well, then mixing your miked sound into your monitor allows you to hear. The only downside to it is you don't want the speaker and mike in a room where people can get at them. Imagine your surprise to pick up a conversation over that feed.
The SM58 makes a good microphone for recording, both live and in the studio. (If you don't like the ball cap on the SM58, just unscrew it, and then it is exactly the same as the previous SM57). Set up the mic on a stand, and position the cap of the mic about 2-3 inches from the mesh that covers the speaker. Many people have their own preferences when it comes to positioning around the speaker, and whenever I mic a stand, I usually place the mic at the 8 or 4 O'clock position.
Where the mic goes in relation to the centre of the speaker is quite important.
To close to the centre of the cone and the mic will record a sound that is too bassy.
Too near the edge of the cone and the mic will record a sound that is too trebley.
What you want to find is the 'sweet spot' you hear engineers talk about. This will of course vary with whatever sound you are trying to record.
When amplifying for live performance, I would suggest placing the mic near the middle of the radius of the cone, and then you can adjust the sound from the Amp's EQ controls.
Hope this helps.
You could probably write a whole book on this subject; certainly people have written whole chapters.
At the end of the day, it's a matter of "what sounds good, works", and electric guitar players have so many different sounds, there's no one-size-fits-all solution.
Based on reading, not personal experience:
Dynamic mics seem to be the basic workhorse of this job. People like SM-57s.
Ribbon mics might sound lovely, if you can afford them, in the studio. They're probably too fragile, and the benefits too subtle, to be suitable for a lot of live work (although, if you have a nice sedate gig in an orchestra pit or something this might work OK).
With one mic, you already have a range of choices. Perpendicular to the cone, you can point it right at the middle, or on the edge of the cone, or any point in between. As you move outward, the sound will get less bassy and more trebly.
Then, you can change the angle of the mic relative to the cone, to get sound from a wider spread of the cone. The only real way is to listen as you go.
You can use two mics in a combination of the positions described above, and blend them together. Try with the mics both in and out of phase (button on the mixer) -- it will make a huge difference, but only you know which you prefer.
Some people like a second mic on the inside of the cabinet, pointing at the back of the cone.
In the studio, you can place a second mic further back, to capture the room ambience. This isn't likely to work live.
Zoog von Rock, crazy frontman of Angelspit, has given a really nice example on how to do this which includes ensuring stereo separation, as well as a balance between low end punch and clarity.
An image to help explain this:
Check out his blog - he does give a lot of info on how they record various things, and how they do the synths they use extensively.