is the mic for live work or recording - or both? The SM57/58 would be ok for live work since it doesn't feed back so easily as some other mics and is very robust. also all sound engineers are familiar with it, so it's a known quantity.
I've found the rode NT3 tends to feedback a bit on gigs and picks up handling noise and thumps and rumbles through the mic stand. The internal 9v pp3 battery lasts for months if you remember to switch it off after use, or it can be used with 48 volt phantom power from a mixer or a separate phantom power supply
I've had useable results from recording with it, but it's a bit disappointing for the price
not tried the rode M3, but it's supposed to be better than the NT3
the AKG C1000 isn't too wonderful, either.
as noted above, the SM57 is better for mic-ing instruments due to it's smaller size and slightly flatter response.
If you're good at electrical stuff it's possible to modify SM57's to make them sound better by changing the internal transformers and messing with the acoustical resonance disc on the front of the capsule, there's lots of info on the 'net if you google "SM57 mods". The info on modifying the resonance disc is here -
http://www.oktavamodshop.com/product_info.php?cPath=3_47&products_id=115
but don't try this if you're not sure what you're doing, or you may end up ruining a perfectly decent mic
Most folks are happy with their SM57's as they are..
there are of course many other mics available, but the SM57 has been around for years and is tried and tested and very reliable, it may not be the best sounding mic in the world, but it certainly works
if you're after higher quality recording mics then i'd have to know a lot more about what equipment you already have and what kind of sound you're trying to achieve and ask advice from some of my friends who do more recording than me. Nowadays I do more editing and mastering on the computer and leave the recording work to the younger folks, so i'm a bit out of touch on all the latest gear
Mr Thomas's info on recording is worth a read -
http://tamingthesaxophone.com/saxophone-recording.html