I'm a website designer in Greenville, SC and have a large number of clients that I've developed for. Most all of these clients will let me put a link to my site at the bottom of their pages. I have a few questions about this:

1. Will it help the my rankings to put links on each client's site back to my site? (most of my clients have content that is completely unrelated to web design)

IF YES:
2. What is the best way to format these links? In the past I've used something simple like "Site Design by YourMark.com, Inc.". Should I be using something with my keywords in it? (a keyword phrase I want to target is "Greenville Web Design")

3. Is it more beneficial to have the link in the footer of all of my client's pages, or just the home page?

Any other thoughts? Thanks in advance.

Use a anchor text for links, include your target keyword or keyword phrase to increase ranks on your target keyword. In forums used a signature that will lead to your site even though its not related, that is the used of signature.

To answer your questions:

1. Yes

2. Yes, but try to stick the name of your business in there as well, for name recognition. I.e.: YourMark.com - the proud product of (a href="yourdomainname.com")Greenville Web Design(/a)

3. I'd go with every page of the site: some internal pages also provide PR creds. If you can - differentiate the blurb.

Thank you, yourmarkscott. You just gave me the perfect idea for a blog post for tomorrow, just when I didn't have any.

This is off topic but...
Since you are targeting the specific region in which you live, be sure to have some vinyl signs made up for your car (your family and friend's cars too, if possible).

For your business customers, see if they will let you put a little sign in front of the business for a few weeks after project completion(like the security systems, pest control, landscapers, etc... do).

Be sure to join you local chamber of commerce, go to meetings and pass out business cards.