Please note, y'all: These prices were gleaned from stores in my local area (Southeastern Michigan) and Internet research. Be sure to ask your local stores what their bag policies are -- These may not be accurate for your regional stores.
Stores that charge you for using their bags (bring your own and you don't have to pay this amount):
IKEA - $.59/bag
Stores that give you a discount, store credit, or something else for bringing in your own bags:
Fred Meyer's - $.05/bag
Fry's - $.05/bag
Kroger - $.03/bag
Safeway - $.03/bag
Sunflower Markets (in the Southwest) - $.10 for every bag of theirs that you bring back and reuse
Target - $.05 / bag
Trader Joe's - Holds a monthly drawing for those who bring in their own bags for a gift certificate.
Whole Foods - $.05/bag for you or as a donation to charity
Special thanks to Frugal Hacks for help with research: http://frugalhacks.com/2009/01/14/stores-that-give-credit-for-bringing-your-own-bags/
Here's to a week of more money in your pocket, and less money lost on bags, peeps!