The technical answer is ZERO. GM has clearly stated that they do not recommend that this vehicle be used for towing.
Now, having said this, I do not understand why. This vehicle (like other "Premium" GM vehicles are cross platform based and the Regal uses all the same frame/power/stability as it's defunct brother the Saab 9-3. Why do you think GM is introducing Turbo engines into the lineup? They're just using Saab turbos and Opal engines in their premium lines (Lecross is the 9-5, Cadillac SRX is the 9-4x etc). The Saab 9-3 was limited not by engine size for it's towing capacity so both the 2.0T and 2.8T versions were capable of towing 3500lbs both on paper, and personally, I've done it with no problem in both 2 and 2.8's.
So, if GM has removed anything from the car then I can say maybe it should be reduced, but saying no capacity at all is crazy. This thing should have no problem towing.
The 2012 Buick Regal has 14.2 cu.ft. of cargo capacity with all of its seats in place.
The maximum tosing capacity of the 2012 Buick Enclave is 4500 lbs..
The maximum tosing capacity of the 2012 Buick Verano is 1000 lbs..
The 2012 Buick Regal has 16 valves.
The 2012 Buick Regal has an inline 4 engine.
The height of the 2012 Buick Regal is 4 ft. 10.4 in. (58.4 in.).
The 2012 Buick Regal's turning circle is 37.4 ft..
The 2012 Buick Regal has double overhead cam (DOHC).
The curb weight of the 2012 Buick Regal is 3600 lbs..
The 2012 Buick Regal is 15 ft. 10.2 in. (190.2 in.) long.
The 2012 Buick Regal has a 6-speed shiftable automatic.
The 2012 Buick Regal has 56.7 in. of front shoulder room.