I second Derrick's choice of the MXL 603. A very inexpensive mic, used to be giveaways on MF bundled with an MXL 2001 large diaphragm condensers.
The 603's are used on pro tour stages for drum overheads, Leslie speakers.

But the lowly SM57 will get you enough signal without sucking in other amps on stage.
After that it is up to your sound-person to figure out the mixing.
For live it will depend on what sound system you are running, the venue, how large a space for you to be away from other amps so noise "rejection" is one factor that weighs in on choosing a mic.
For studio you would be better off leaving that to the studio as they have their mic cabinets and styles they like to use. Isolation will be ironed out for you to be able to use great mics. And many times those mics go into the multi-thousand dollar range and don't do well travelling for live tours.
The all-around SM57 is still used on pro stages. But your preferences should weigh in and you would need to know how to use your mic of choice or like what your sound person is doing with yours or his/her choice of mic for your amp.
It's not as simple as choosing a mic and thinking the mic itself will answer all the problems of you being heard. The mic has to address more than one issue.
You could spend a lot of money on a mic and then get turned down in the mix to where it sounds like you're running through an SM57 or your mic being turned off altogether unbeknownst to you because your instrument is showing up in the singer's and some other mics including your amp's speakers beaming into the audience.
Basically what you think your instrument should sound like does not always equal the way it gets represented out front in a live situation.
For studio this would work out similar. It will depend on the engineer and producer's idea of what role your instrument plays on a track not unless it's a steel album starring you.