What are the failure rates for piston, turbine, and jet engines? (If more subdivisions of "engine" are necessary to make the data useful, please point them out).
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.
The statistics above are very misleading as to failure rates of turbine engines. Keep in mind ETOPS engine fail rates are based on shutdowns. Most of the time, the engine does not fail but is shut down as a precaution because the aircraft has sensed some deviation from normal engine parameters. Often this turns out to be a fault in the monitoring system as opposed to an actual failure.
I do not have the statistics at hand but I can say with some certainty that the failure rates of piston engines are tens of times higher and the fatality rates are much higher as well.
I just retired from airline flying with close to 30,000 hours. 26,000 of those were in jets and jetprops and I NEVER had an engine failure or a precautionary shutdown. In the 4,000 hours of piston time, I have had four engine failures, all in twins. Two were complete and two were partial. I have owned 4 different private planes. Two piston singles, a piston twin and now fly a single engine turbine powered aircraft.