Will it work to just plug a Shure SM-58 into the iPhone 5?

No extra software, no special mic, the 58 is the most common mic out there, and I have one. If I just get any old xlr to 1/8" adapter, and run the mic right into the headphone jack, will it use the mic to record the audio while I'm recording video?

That's what I'm trying to do, I've Googled around, but there just seems to be a lot of talk about how some mics work and some don't, and it's not clear if you need a special cable.

There are two issues to deal with in what you want to do. Dealing with your second mention first, yes, you do need a special cable/adapter - an ordinary XLR to 3.5mm mono, or even stereo, will not work. This is because the headphone jack on an iPhone is not only a Headphone Out, but a Mic In, which means it needs an extra conductor. Look closely at ordinary stereo headphone plugs and you'll see TRS (Tip Ring, Sleeve) - 3 conductors for the two (Left/Right) signals and a common return. But the headphone plug you have for your iPhone is TRRS (an extra ring). Adapters are made to handle this - here is one to go from XLR to iPhone jack:

http://www.kvconnection.com/product-p/km-iphone-xtrs.htm
but it's almost $32 (before shipping) - here are instructions for cobbling together your own adapter for about $10 (about 2/3 down the page)
http://blog.watch.tv/2011/11/3559/

The other issue (and potentially more difficult to overcome) is impedance matching. Your Shure just may be too good a mic. It's rated at 150 Ohms impedance and is looking for a sturdy preamp ahead of a PA amplifier or mixer - see specs here:
http://cdn.shure.com/specification_sheet/upload/82/us_pro_sm58_specsheet.pdf
I honestly don't know whether or not it will put out enough usable signal for your iPhone to use. Maybe someone with that knowledge can chime in on this part. I, like you, have seen both yes and no answers online and have no personal experience. Here are a couple videos that talk about this:

I'm sure it would work with a "powered mic", but your Shure SM-58? It should work, but perhaps not a high enough signal-to-noise ratio to be useful. Hope this helps some… Good luck!