What simulation software do airlines use?
X-Plane offers a "Professional Level" which is mostly about licensing. However X-Plane also has an FAA certified version that if paired with proper controls is legal for certain training/instructional hours. You can find more info on the FAA's approved simulators here and here. The flight school I trained at had a certified sim from the 80's that was pretty miserable by todays standards. It was certified and had no screen, I would say it was "custom built" FWIW.   

Many airlines may turn back to the manufacturer for recurrent training. Both Boeing and Airbus run facilities with full motion simulators. I would think they built the software in-house but I don't have any hard evidence on that. As the OEM builder they have access to genuine interior parts to build the sims.     

For smaller aircraft places like SIMCOM offer training in full motion sims but do not list their software of choice. SIMCOM uses actual aircraft interiors (most likely from defunct airframes) for their sims. 

If your really adventurous you can build a full cockpit sim yourself.