That expression "Once in a Blue Moon" is usually used to express a rare event. However, since Blue Moons are separated by roughly 2½ years, relative to our lifetimes they are not really that "rare." Sticking to basic full-moon-phase related events (eclipses excluded), there are two other examples which better exemplify rare occurrences. One is when February (only February) does not have a full moon. The time between full moons (lunation) ≈29.53 days which is always greater than February's length, but because the lunation is not exactly resonant with our days, when and how often Feb is moonless is complicated. A second example is when a full moon occurs on Leap Year (Feb 29). Now how often is that!?
I looked at all full moons over 2000 years (1000 AD to 3000 AD) to get an answer about these rarer occurrences. For convenience, I call a moonless February a Red Moon, and a full-moon Leap Year a Leap Moon. In that time span, there are/were 827 Blue Moons, 91 Red Moons, and 18 Leap Moons. The average time between each like-moon is 2.42 years, 22.0 years, and 108 years respectively. (Note: The fact there are Red Moons explain the primary deviation from 7 moons every 19 years for a Blue Moon; on average, every 22 years there is an extra Blue Moon). The distributions of these events are visible in the graphs.
- Blue, Red and Leap Moons - 1950 AD to 2050 AD
(Universal Time)
- Blue, Red and Leap Moons - 1000 AD to 3000 AD
(Universal Time)
You can use these graphs in conjunction with the NASA Catalog to see the details.