If it's a good match to the voice, then it's a great mic for recording vocals. It has a high-mid frequency bump (somewhere between 3 & 4 khz? I think), and I believe that's intended to make a voice cut through a live band mix, and it may sound harsh or weird in an isolated setting, but if not (because maybe the voice benefits from a boost in that eq region), then go for it.
If I could only own 1 microphone it would be an SM58.
Use it for Live vocal, studio vocal, guitar amps, drums (all), brass, everything really. Not the best at all things but will do a good job of all things.
This might be because a more dynamic (read: louder, less controlled) performance can be tracked easier with a dynamic mic. In other words: dynamics are good for screamers or people who haven't developed mic technique yet on a condenser.
Antichef is right on the money though - a 58 can be great for some voices and bad for others. I've tracked rock bands in the studio live with a 58 on vocals before I got a SM7. Depending on the singer it works.
And it can def. be used on cabs, toms, snare etc. Pretty hard to beat for the price. But check out Sennheiser & Audix too if you're looking for vocals alone - some singers sound crap on a 58 and great on a Sennheiser 835.
It's not about "picking up from the chest," it's about two different types of microphones and the way they work. Sometimes a condensor (like your Rode) is the right mic for a certain voice. Sometimes a dynamic (like the SM58) is.
That being said, I own both the NT1A and a Beta 58 (among many other mics). If I was limited to just those two, the 58 would be the one I'd try first.
If the singers voice is too boomy with a 58, tell him to stop swallowing it and to back away from it an inch or more.
If holding onto the mic, don't let him hold it with his hand wrapped around the back, if you hold a 58 close to the back with your fingers on the back of the ball it changes the sound completely and makes it very muddy.