Which is more rare: Lunar eclipse or Solar eclipse?
Solar and lunar eclipses occur about equally often - between two and four times per year.
However, when you do not intentionally travel around the world chasing solar eclipses, you are more likely to observe more lunar eclipses. The reason is that solar eclipses can only be observed from a comparatively small area while lunar eclipses can be observed from anywhere were the moon is visible when they occur, which is half the planet.
They are not that rare; there are 4 to 6 solar and lunar eclipses each year with an equal number of each (on average).
I think the OP means to ask which phenomenon is more *frequent* and *why*. Could you add a (link to) an explanation as to *why* they are equally frequent?
The answer is "it depends." It depends on what kinds of eclipses you count, and perhaps even more importantly, how likely you are to see the event.
There are four types of solar eclipses:
Total solar eclipses, in which the Moon blocks the entirety of the Sun at some point on the surface of the Earth. About 26.7% of the solar eclipses are total eclipses.
Annular solar eclipses, in which the center of the Moon as viewed from the passes close to the center of the Sun, leaving an annular ring of sunlight around the Moon. About 33.2% of the solar eclipses are annular eclipses.
Hybrid solar eclipses, in which the eclipse changes from total to annular (or vice versa) along the path of the eclipse. About 4.8% of the solar eclipses are hybrid eclipses.
Partial solar eclipses, in which no part of the Earth sees a total or annular eclipse but some part of the Earth does see the Moon as partially blocking a portion of the Sun. About 35.3% of the solar eclipses are partial eclipses.
(Source for numbers: http://eclipse.gsfc.nasa.gov/SEcat5/SEcatalog.html)
There are three types of lunar eclipses:
Total lunar eclipses, in which the entirety of the Moon receives no sunlight because the Earth is between the Moon and the Sun. About 28.8% of the lunar eclipses are total eclipses.
Partial lunar eclipses, in which the part of the Moon receives no sunlight because the Earth is between the Moon and the Sun, but there is no time at which all of the Moon receives no sunlight. About 34.9% of the lunar eclipses are partial eclipses.
Penumbral lunar eclipses, in which the part of the Moon sees the Sun as partially eclipsed by the Earth, but no part of the Moon sees the Sun as totally eclipsed by the Earth. About 36.3% of the lunar eclipses are penumbral eclipses.
(Source for numbers: http://eclipse.gsfc.nasa.gov/LEcat5/LEcatalog.html)
From the above two sources, there have been / will be 12064 lunar eclipses and 11898 solar eclipses between 2000 BCE and 3000 CE. These are essentially the same number. In this sense, solar eclipses and lunar eclipses are equally likely.
With regard to the likelihood of your seeing an eclipse, lunar eclipses are much more likely than solar eclipses. For any given lunar eclipse, a bit over half the world gets to see at least part of the event. For any given solar eclipse, only a tiny portion of the world gets to see part of the event, and for most of them, it's just a partial dimming.
The reason that solar and lunar eclipses are about equally common, is that they occur when the moon is "close enough" to the plane of the ecliptic at the point where it is full (lunar eclipse) or new (solar eclipse). Otherwise it completely misses the sun (solar eclipse) or the earth's shadow (lunar eclipse), and you don't get even a partial eclipse.
If the moon orbited precisely in the ecliptic plane, then there would be a total lunar eclipse every full moon, and every new moon a solar eclipse that's either total or annular according to the distance of the moon from the earth at that moment. But the moon doesn't orbit in that plane, it intersects it twice a month, at a time that most months isn't when it's close enough to being lined up with the earth and sun.
"Close enough" in the case of a solar eclipse is when the earth, as viewed from somewhere on the sun, is partly obscured by the moon (therefore: part of the sun is obscured by the moon when viewed from part of the earth). Whereas "close enough" for a lunar eclipse is when the moon is partly obscured by the earth from the somewhere on the sun (and therefore part of the earth's shadow falls on part of the moon). This is just about the same size target area.
What the sun "sees" as the moon orbits the earth is similar to what we see when we look at the moons of Jupiter: the moon moves from one side to the other and back. Sometimes it passes "over" the earth, sometimes "under", sometimes through. What's required for a partial eclipse of either kind is that it moves through, such that those two disks intersect as viewed from the sun. The difference from Jupiter, is that Jupiter is really big, so passing through is more likely for the moons of Jupiter than our moon.
If you really want to know which kind is a few percentage points more or less frequent, then you're beyond my ability to predict. Consult a table of eclipses covering a long period of time, or wait for an answer from someone with better qualifications :-)