White:
From Argentina, Bianchi New Age White (a 50/50 blend of Malvasia and Sauvignon Blanc). Off dry, fruity, and aromatic. I tried this at an industry tasting last week, and it was one of the more interesting $10 whites I've had in a while. Importer:
http://www.quintessentialwines.com
An Argentinian Torrontes would also be an interesting option. If you want something French, Gascony produces cheap whites of wildly varied quality (some quite good, some equivalent to jug wine).
Santa Rita 120 is an option for inexpensive, good for industrial-scale whites and reds from Chile.
Red:
The Spanish reds that others have suggested are good ideas, but I'd want to save Spain for the sparkling wine.
Want a $10 merlot that is actually interesting? Try this from Chile: Santa Ema 2003 Reserve Merlot (another one from the same tasting). Importer:
http://www.wineofakind.com
Other options:
Boutari Kretikos Red (Greece) -
http://www.paternoimports.com
Fairview Goats Do Roam Red (South Africa) -
http://www.goatsdoroam.com
Italy also has some inexpensive reds. How about a Primitivo (aka Zinfandel)?
Rose:
Lots of dry roses from Rhone/Provence regions of France.
For something more interesting, try this from Italy for about $10: Antichi Vigneti del Cantalupo 2005 ""Il Mimo" Nebbiolo Rose Colline Novarese DOC, sorry no national importer information for you. In Chicago, it's imported directly by Maverick, a local distributor.
Sparkling:
Spanish Cava is your best bet for inexpensive sparklers.
For something a little more off the wall, how about a demi-sec from South Africa? Pearly Bay Celebration Sparkling. Importer:
http://www.57mainstreet.com
Let us all know what you end up selecting and what people thought of them.