How common is engine failure?
Engine failure is not that common. For a gas turbine engine, a press release by GE gives some data as a reference point. In 1995 they quoted various engines as having a dispatch rate of between 99.99% and 99.89%. That is only 0.01 to 0.11 % of flights could not depart due to an engine issue. However, that doesn't identify in flight failures, best captured by the inflight shut down (IFSD) rate. Data on this is difficult to get for specific engines, as the OEMs consider it competitive information, so the best guide is probably as mentioned the ETOPS rating. The ETOPs fleet is reported as achieving a shut down rate of less than 0.01 per 1000 engine hours. Another way of defining reliability is the (repair) shop visit rate (SVR). Figures given for this (by GE in the same link as above) were between 0.085 and 0.231 visits per 1000 flight hours. However, this covers both scheduled and unscheduled maintenance (i.e. failures), so perhaps doesn't identify what you want.

For piston engines, the answer to this  question probably has some useful data. In addition, Continental claimed in 2014 that they had an IFSD rate of 0.0061 per 1000 flight hours, which they claim is the most reliable piston engine in general aviation. This compares with "less than" 0.01 for gas turbines, so, it seems very similar. 

It maybe possible to find a better value for the IFSD rate for gas turbines. Given the rate is so low, it really needs to be quoted to more than 2 decimal places, or given per 100,000 flight hours. Ultimately, I would expect the IFSD rate for gas turbines to be lower than for piston engines, due to the huge economic incentive gas turbine manufacturers and the commercial airline industry has increasing reliability for turbofan engines, that does not exist to the same extent, for general aviation piston engines.