As an engineer I have been involved in two.
The most recent a 757/RB211 tail pipe fire occured as a result of low APU duct pressure and a slight tail wind. The engine suffered a hung start and following shut down, I went out and had a chat with the crew via the intercom. As I was talking I noticed smoke coming out of the front of the right engine and advised there may be a tail pipe fire. The captain confirmed rising EGT and initiated a dry motor while I went to have a look. After an age the captain said that he had to stop as starter duty cycle was at the limit. but the thing was still burning slightly. Just as I was about to get the fire crews to put it out with some CO2 it went out of its own accord. A quick inspection and confirmation that EGT had not been exceeded and it was on its way (after an ECU reset to restore duct pressure)
The other many moons ago was much more exciting.
The a/c was a 707 JT3 and the tail pipe fire occured after an engine run following replacement. On being advised of the fire the usual dry motor was carried out but the fire was still huge at end of the duty cycle so the only option was to light her up again
The torching that occured as this was done was MOST impressive! Having then idled the engine for 10 minutes a normal shutdown was sucessful.
Do not be confused with torching and tail pipe fires. Torching, particularly after installation of an engine that has been inhibited, is quite common. There is a famous video of a 737/CFM56 which has 20-30 foot flames coming out of the back as it is started for the first time probably because it was inhibited. This causes a delayed light-up and sometimes some spectacular displays.