What is the current capability of the common "C" wire for house thermostats
The only wire currently sold by US big-box stores for thermostat use appears to be AWG18 x whatever number of conductors. 

Most of the transformers are 20-40VA (some are 10VA though) and are so marked, along with Class 2/Class 3. Many are impedance-protected as well as fused so they may just limit the current and get hot if you short the output (and if a primary fuse does not blow). They probably have terrible regulation as a result. 

Anyway, a 10VA transformer would be be able to supply about 400mA RMS at 24VAC or if you full-wave rectify it and filter it about 250mA at 30VDC or so. Scale up for the VA of the transformer. 

If you do test the short-circuit current of the transformer you might want to find out where the primary is fused and make sure you have an equivalent spare on hand, because the primary fuse (often something like a glass or ceramic fuse on a furnace control printed circuit board behind a service panel on the furnace) may well go. It won't blow a mains circuit breaker, more likely a somewhat hidden fuse. 

Note that it's normal to have the supply for the central air control in the furnace and if you want to remove power from the 24V and the 240VAC main power you have to turn both off. 

Also, keep in mind that the 'C' wire is not always available at the thermostat location, especially in older systems. For that reason smart thermostats often have a battery inside and may cycle the heat or cool to keep the battery charged, even if unneeded-- this is so they can have the largest possible market. Homeowners may be fine with rewiring an existing thermostat, but fishing a new wire through various finished interior walls is a bridge too far.