I've been using the same Shure BG 5.0 condenser mic for over 20 years, recording mostly my own vocals. I also own a beloved Zoom H1 portable stereo recorder, which I use for field recordings. I don't believe I can plug the Zoom into my mixer, and anyway, I'm pretty sure one of the two mic inputs on the mixer is shot, so I couldn't anyway. (I will record onto the Zoom and import the resulting wav into Sonar.)
 
I am going to have some people come in and record in my home studio. They are simple solo guitar and solo vocal. I'd like to get good recordings of them, and do some mixing later on, possibly adding some other instruments if inspiration strikes. I want to get a usable raw recording of the performance, but these aren't the kind of people who are going to do well recording guitar and vocals in separate takes. I need to record their complete performance and then mix and balance later. 
 
So I was thinking I could put one mic close to the guitar and another close to the vocal. There will be plenty of bleed, but it will give me at least a little balance and mix capability later. My Shure is mono and the Zoom is stereo; also, I don't know really anything about different mic types. Lastly, I do not have a stand for the Zoom, only the Shure.
 
Should I record the vocals stereo or the guitar? How would you record this?
 
Non-negotiable conditions:
* Shure Condenser mic (mono)
* Zoom H1 portable recorder (stereo)
* Performances are single take of guitar and vocal
 

Hmmm. "Should" is one of those words I try to avoid because it implies obligation and that the other choice is inherently wrong. I have experience with both mikes. The Shure is very flexible and imo has a better overall quality. My tendency in a live setup is to give vocal best mic. Much easier to cover recording quality issues on guitar than vocals. Reverb, chorus and delay on mediocre quality guitar recording may pass as planned but voices drownedb in such are far more noticable. The zoom mic is decent. A stand woild certainly help but again, I defer to giving vocals first preference.

I agree with Dub - record the guitar in stereo and use the Shure for the vocal. I'd definitely look around for a stand for the Zoom because you're going to want to experiment with different positions. Make a few short test recordings with just the guitar to determine the optimal place to position the microphone(s). Check Craigslist for a cheap second-hand mic stand. If you don't have the adapter for the H1 you can use a heavy-duty rubber band to attach the recorder. Wrap the stand with a dishrag first so the H1 isn't touching the stand directly.
 
Locate the performers as far away from one another as they're comfortable, which might only be 6 feet but the greater the separation the fewer phase problems you'll have. Have them face one another so that the vocalist's microphone is facing away from the guitar and the dead zone of the cardiod pattern is facing the guitar.
 
Beyond that, just make sure nothing's getting distorted going in. After the first take, tell everybody to shut up and then play it back through headphones to verify. Have fun!

+1 bitflipper.