Looking at the numbers from Wikipedia's comparison of orbital launcher families:
- RD-107 family (R-7 rockets, including the Soyuz): approximately 9800
- RD-253 family (Proton): approximately 2600
- YF-20 family (Long March 2/3/4): approximately 2300
- RD-0210 family (Proton upper stage): approximately 2200
- RD-0110 family (R-7 upper stage): approximately 1900
- LR-89 (Atlas family): approximately 1000
- Merlin family (Falcon rockets): approximately 950
- Viking family (Ariane 1-4): approximately 750
- RD-215 (Kosmos rockets): approximately 600
- LR-105 (Atlas family): approximately 500
- RD-250 family (Tsyklon rockets): approximately 500
- 11D49 (Kosmos upper stage): more than 450
- LR-87 family (Titan rockets): approximately 370
- LR-79 family (Thor rockets): at least 350
- H-1/RS-27 family (Saturn I, Delta II): approximately 350
- Rutherford (Electron): approximately 350
- RD-170 family (Energia, Zenit, Atlas V): approximately 200
- RS-56 (Atlas II): approximately 200
- Vulcain family (Ariane 5/6): approximately 120
The number of engines drops off fast after this, with around a hundred engine families with fewer than a hundred examples built.
Two major omissions from this list are the RL10 family (used in the ubiquitous Centaur upper stage) and the AJ10 family (used in many upper stages, including Delta II, Titan III, the Apollo service module, and the Space Shuttle OMS engines). Since both of these engines are used across a wide range of launch vehicles, it's unclear how many of each has been built. It's likely that the RL10 sits somewhere between the RD-170 and the Rutherford, while the AJ10 sits somewhere between the RD-250 and the Merlin.
Also omitted from this list are vernier and RCS engines, solid-fuel engines, and upper-stage engines from rockets that can mix-and-match upper stages.