
Shure
SM58
Indestructible live workhorse; but muffled for studio vocals.

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Lots of cheaper mics that IMO sound better than the ol' 58. Samson Q series, Berhringer. The Sennheiser e35 is a better mic for the money. The Shure PG48A and 58beta are also great.
try super cardiod mics like the se v7 or even shure beta 58a. have your drummer tape his cymbals. turn guitars and bass down on stage. use in ear monitor. project your voice more but not like you are doing to hurt yourself etc
Feamle vocal in our band bought SE V7 while I use 58A beta but that mic is much worse than my spare SM58 and we use that one rather than hers because it sounds much worse and it needs so much tweeking to do anything.
beta58 has weird handling noise
Sm58a is technically a handheld
It sounds like you only have 1 mic/1 input available at any given time. The part I don't like about that is that you might be missing out on the possibility of a stereo image on the piano. If you were to ask me(and probably ONLY me), I'd say 2 of those Beta 58s or even the Beta 87s would be just fine on a piano, BUT YES... it WOULD pick up a lot of the clack and clunk from the keys.... however, some people like that. Even if you only had 1 mic for the piano, those Betas sound pretty good. If you are looking for an upgrade on mics that will give you a studio sound with less "condenser" sensitivity, I would recommend Ribbon Mics like the sE Electronics X1R. These mics are not too expensive at all, and they sound like a dream (especially for how inexpensive they are). However, they are definitely not as "bright" as a condenser mic. The other thing I'd say is that even a standard SM58 sounds decent for vocals in a studio setting, however if it is true that you only have the one input, then regardless of mic choice; I'd recommend recording the piano first AND THEN the vocals. I certainly would not recommend using a single mic to record both the vocals AND the piano at the same time. As for the other mics you mentioned... The SM7b is AWESOME on female vocals, but it IS pretty pricey even for how good it sounds(at least in my opinion) and the Rode NT1 is simply a horrible choice all together. I'd rather one of the Betas on a piano before the NT1. You'd definitely be better off with a Ribbon mic (or 2, since you should be going for stereo image anyway) OR even an NT5 before that NT1. In any case.... Those fluid performances CAN still be recorded independently from each other, especially if you are like me because I cannot fathom doing an entire performance without a click for at least the main portion of it.
Shure SM58 or Beta 58. Lovely, rugged, clean-sounding dynamic mics that excel at off-axis rejection and maximizing S/N ratio. The Beta 58 has a slightly higher output level; the SM 58 does slightly better at reproducing bass. You won't be unhappy with either!

Shure
SM58
Indestructible live workhorse; but muffled for studio vocals.

Shure
SM57-LC
Versatile for instruments; needs pop filter for vocals.

DJI
Mic 2
Unbeatable convenience, 32-bit float; but poor Bluetooth audio.

Samson Technologies
Q2U USB/XLR Dynamic Microphone with Accessories
Great value dual USB/XLR; recent price increase noted.

Shure
SM7B Vocal Dynamic Microphone
Professional vocal sound; but expensive, needs external amplifier.