Where can I find the cheapest sign and drive lease currently in the market?

THe economy isn't great right now, but if you're in the market for a car, now's a great time to spend some money.
Companies with great lease deals right now include Acura, Honda and Lexus. Additionally, General Motors is offering several lease deals and rebate offers that will probably help you make your decision.
 
I'd recommend looking into leasing one of the GM vehicles, such as a Chevy Cobalt or the Aveo. Both cars sell new for around $13,000, so a lease would be very reasonable on either car.

Lease figures will vary upon location and dealer.

It's funny... because I have been recently looking into this because I really want to lease a car.  I checked everywhere for the best deals where the monthly payment is cheap, the car is good, and the deposit is minimum.  I want to note that a lot of people don't realize leases are negotiable.  So when they say you can only limit to 10,000 miles / year, you can EASILY bargain up to 15k, but not above that.  Best deal I could find right now is a Honda 2009 Civic Coupe... it's about 149/ month, and the deposit is around 2000.  However, there's another option where you can spread out the 2000 deposit throughout the monthly payments, i.e.  pay 300/ month instead of 150, but you don't have to put down.  Good luck!   Also consider sites like http://www.swapalease.com or http://www.leasetrader.com, where you take over people's leases and also you don't have to put down that deposit =).
Don't bother with Lexus or Acura... it's real expensive unless you have money.