The low-mass X-ray binary 4U 0614+091 is a source of sporadic thermonuclear (type I) X-ray bursts. We find bursts with a wide variety of characteristics in serendipitous wide-field X-ray observations by the WATCH on EURECA, the ASM on RXTE, the WFCs on BeppoSAX, the FREGATE on HETE-2, the IBIS/ISGRI on INTEGRAL, and the BAT on Swift, as well as pointed observations with the PCA and HEXTE on RXTE. Most of the bursts are bright, i.e., they reach a peak flux of about 15 Crab, but a few are weak and only reach a peak flux below a Crab. One of the bursts shows a very strong photospheric radius-expansion phase. This allows us to evaluate the distance to the source, which we estimate to be 3.2 kpc. The burst durations vary generally from about 10 s to 5 min. However, after one of the intermediate-duration bursts, a faint tail is seen to at least about 2.4 h after the start of the burst. One very long burst was observed, which lasted for several hours. This superburst candidate was followed by a normal type-I burst only 19 days later. This is, to our knowledge, the shortest burst-quench time among the superbursters. The observation of a superburst in this system is difficult to reconcile if the system is accreting at about 1% of the Eddington limit. We describe the burst properties in relation to the persistent emission. No strong correlations are apparent, except that the intermediate-duration bursts occurred when 4U 0614+091's persistent emission was lowest and calm, and when bursts were infrequent (on average roughly one every month to 3 months). The average burst rate increased significantly after this period. The maximum average burst recurrence rate is about once every week to 2 weeks. The burst behaviour may be partly understood if there is at least an appreciable amount of helium present in the accreted material from the donor star. If the system is an ultra-compact X-ray binary with a CO white-dwarf donor, as has been suggested, this is unexpected. If the bursts are powered by helium, we find that the energy production per accumulated mass is about 2.5 times less than expected for pure helium matter.