What are the reliability statistics for modern jet engines?
What is the actual numerical reliability statistics of the modern, large-aircraft, commercial turbofan engines available today?
e.g. How many running hours can we expect a failure after? Are there empirical or projected MTBF statistics?

Buried in this GE press release is the comment that the GE90 has 'only one engine IFSD  per one million engine flight-hours'.
This ICAO presentation has a lot more info, including these threshholds for ETOPS IFSD rates:

Propulsion system should achieve an in-flight shutdown (IFSD) rate of:

0.05 / 1000 engine hours for EDTO 120 min
0.02 / 1000 engine hours for EDTO 180 min
0.01 / 1000 engine hours for EDTO beyond 180 min
Rate computed based on world fleet data, on a 12-month rolling average


This strongly implies that the reliability targets are empirical, not based on theoretical numbers.

Is there a significant difference in the failure rate between engine models or manufacturers?

Page 56 of the presentation above has some data, but IMHO not enough to draw any conclusions.

How rapidly has engine reliability improved? Do we have a time trend of reliability numbers?

All I can say is it's definitely improved massively, but I'm not sure where to find the info.