I'm considering purchasing a Shure SM58 for use both in recording and when busking (street performing). When doing the latter I use iReal Pro on my Android phone as a backing band, and have a UE Boom Bluetooth speaker for amplification. What's the best way to use a decent quality vocal mic in this context? A Bluetooth mic would be much more convenient but I'm not sure if there are Bluetooth mics available with decent sound quality. Otherwise, are there XLR-to-bluetooth adaptors available, and if so, would they provide better sound quality (e.g. gain level) than I would get by using an XLR-to-1/8" adaptor to connect the cable to the headset jack on the phone?
I don't know your specific equipment [other than the good ol' SM58] however, what you will rather rapidly discover is that you cannot use Bluetooth in any synchronised audio throughput scenario.
However you get the mic into your system, you will have latency.
If your input & output are being controlled by the same structure, it might be possible to limit that input latency to not make it unusable, so long as you are hard-wired.
However, if you are going to then run that output over bluetooth, then your output vocal is going to be so far delayed compared to when you actually sing, that you will not be able to work with it.
These latencies can be combatted if you are recording - the software can measure & compensate 'to tape' for later playback; but live, you haven't a hope, sorry.
Bluetooth would work for a backing track. It does have latency, but you'll automatically accommodate it.
Once you start introducing instruments that transmit over bluetooth, you're going to get delays that don't match up.
My suggestion (only one solution - there are many out there) would be to get a small 5-channel mixer. Feed the phone into one channel (use a 1/8" stereo ->2 x mono RCA, or 1/8" Stereo-> 2x1/4" Mono, cable - depends on the mixer). Bring the SM58 in with a standard XLR cable. Take the output from the mixer, and feed it to the speaker (you'll need a 1/4"->1/8" Stereo cable, or XLR->1/8" cable, depending on the mixer). Buy the mixer at a local pro music shop, and take your busking stuff with you - they'll make sure you get the cables you need.