I can’t comment on how unusual they are in general, but one can certainly find plenty of zero-inflated continuous dists in insurance claims severity data, and probably anywhere there’s conditional outcomes on events.
McElreath has a nice example here http://xcelab.net/rmpubs/Mcelreath%20Koster%202014.pdf for a zero-inflated gamma likelihood on the quantity of meat returned from hunting expeditions. I used this model principle with success for a zero-inflated lognormal likelihood.