At CAL we do have and use FOQA. The company pulls the data from the DFDR of each aircraft on a monthly basis and sends it directly to the union for de-identification and analysis. The union will help the company to spot trends and if a particular flight or pilot is seen way out of normal parameters, the pilot(s) are normally called by the union and asked to provide further info. The company does not get any of this info for punitive or disciplinary purposes.
In addition to this program, we also have the same data displayable through the ACARS system after each leg that shows multiple parameters such as stability at 1000, 500, and 100, when the gear was lowered and at what speed, flap extension speeds, etc. This will also give a pitch report for the pilot to view that shows the proximity of the tail to the ground on takeoff and landing.