An even better way to save money buying used gear is to search for deals on "broken" or non-functioning gear. Sometimes you can discern from the product description what might have gone wrong with the microphone, but other times you just have to take a leap of faith and hope that the damage is something that can be fixed. This assumes you have a basic working knowledge of how to use a soldering iron-- if not check on the web for some tutorials about soldering, and practice first on a non-critical piece of gear to hone your skills before moving on to a mic.
This how-to specifically applies to a Shure SM58 or SM57 which I have found to be one of the more difficult dynamic microphones to fix because of the delicate construction of the mic. The same repair concepts apply to pretty much any dynamic mic, and may be actually easier to implement with other models and brands, depending on the housing, wiring and microphone capsule. (Condenser microphones are infinitely more difficult to repair. Since a condenser mic element is especially fragile and more susceptible to vibrations and shocks, any damage might be beyond repair; purchase broken condensers with caution.)
First off, with a non-functioning microphone you have to determine the most likely area of malfunction. Carefully unscrew all the pieces and inspect them for wires that are not connected or loose. Most frequently, the cause of a non-working microphone happens when a wire disconnects and breaks the signal flow-- in this case a quick re-soldering job will get you up and running in no time. If the wires are still connected toward the front of the microphone, find a small flathead screwdriver and unscrew the XLR connection from the base of the mic and remove with pliers.
The photo left shows the most typical problem with mics. One of the connecting wires to the XLR has been disconnected, which can be easily reattached with a touch from a hot soldering tip. If more than one wire is disconnected and you're not sure which wire goes where, try to find some documentation from the manufacturer to give you more information on the original wiring setup. (Note: the XLR ends on the SM58 at left don't match Shure's wiring diagram. Shure microphones have gone through several wiring permutations over the years, and this particular mic is over 20 years old, so for whatever reason has 4 wires instead of 2 or 3.) If it is in fact a wire that's come loose, you're lucky; this is the easiest fix and makes you feel like a bandit from all the money you saved on a "broken" mic.The more tricky fix comes into play if there's no visible signs of disconnected wires at the XLR end or in the middle of the mic. If not, there's probably been some kind of damage to the wiring of the microphone capsule-- the type of damage will determine if you can fix the mic or not. You'll have to carefully disassemble the microphone capsule end from the wires on the base of the 58, first by de-soldering the base wires from the metal terminals at the end of the capsule and unscrewing the center nut and any ground wire (if present). Then you'll have to slide the entire capsule element out of the microphone housing, making sure not to cause any further damage to the capsule from the force of the disassembly. Removing the capsule from the housing will take some nudging and wiggling to get it out, and once removed you might not be able to get it back into the housing as snug as it was before.
When de-soldering and taking apart the capsule element, it would probably be best to take pictures of how the mic was assembled for reference, and clearly mark exactly how the wires connect to the capsule and the diaphragm. When reassembling the microphone, you want to make sure the correct wires are connected back to their origins, otherwise you will reverse the polarity of the mic, and it won't function the way it should (although you'll still get a signal).
Once taken apart, you'll be able to see where the damage has occurred to the microphone capsule. The most frustrating part of a 58 or 57's construction is the delicate wire that leads from the coil under the diaphragm, between strips of yellow tape, down the magnet of the capsule, and finally connects to metal terminals secured on a wafer on the center post of the capsule. Dynamic Shure microphones are extremely durable, but if anything is really going to "break" on these mics, it's going to happen in the few exposed centimeters where the delicate red wire connects to the metal terminals. Fixing this requires near-surgical precision and a steady hand.If indeed the little red wire has become detached from the metal terminal, you'll have a hard time soldering it back on. The red coloring is actually a type of plastic shielding, so solder won't readily adhere to the wire without some prep work. First, carefully peel back the top layer of the yellow tape with an Xacto knife, being careful not to rip the red wire in the process. With some electrical tape, secure the top layer of yellow tape out of the way, and separate the red wire from the bottom piece of yellow tape. With a lighter, move the flame close to the wire, making sure to never touch the wire directly, and only exposing the wire to heat for a very brief instant. This will burn the shielding off of the wire and leave the wire exposed, but if the flame is too close, the wire will burn, and then it's probably game over.
Now the tiny wire is probably too brittle to reattach to the original terminals, so the next step is to manufacture an alternative method that will leave the tiny wire securely in place while allowing you to make a connection with the colored wires from the base of the microphone. You'll need some copper desoldering braid, some soldering flux and the rest of your soldering tools.With the desoldering braid, cut a piece the length of the capsule magnet and poke a hole with a pushpin through the braid.
Thread the tiny wire through the piece of braid and tape the braid on top of the yellow tape to hold it temporarily in place. Put a little bit of soldering flux over the hole where the tiny wire is threaded-- this will help ensure a good bond between the braid, the wire and the solder.Next, solder over the hole, filling it in and letting the solder run down a portion of the braid. Don't use too much solder, but make sure that you've used enough solder to make a solid connection with the tiny wire. It's OK if not all portions of the wire adhere to the braid-- as long as you've made a
connection somewhere along the braid, it's probably sufficient to pass signal from one element to the other.Once the wire and the braid are soldered, lay the top piece of yellow tape back over the braid and secure the whole thing in place with a couple of wraps of electrical tape. At the bottom of the mic capsule, solder a spare piece of wire to the braid-- this new wire will jump the gap between the mic capsule and the base of the microphone. Once again, be sure to note which side of the microphone corresponds to which color of wire, otherwise you run the risk of reversing your polarity.
Inserting the capsule back into the housing is a difficult squeeze since there are several new layers of tape, but with enough care, the capsule should eventually fit into place. Twist the extension wires with the original wires from the base and solder the connections, then shield off the wire with a strip of tape. Screw the top portion of the mic back into the base and test the connection-- if done correctly, you should have a functioning microphone once again.*Edit* Check out Shure's documentation for the SM57 here and the SM58 here. Here's a quick wiring diagram provided in the SM57 manual:

