A few days ago, I typed Yoshi in to a Google search, and looked at pictures for fun; most of the Yoshis were green.
Are green Yoshis more common than other varieties, or are they all equally common, with green ones just being the most well known?
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In context of TV and Movies, only a green Yoshi has appeared in the Super Mario Bros TV Show. You can not get more common than "the only one available". The same goes for the Super Mario Bros. movie, although there is a clear difference in the appearance of the character due to the nature of the movie.
That being said, the TV Show is very obviously based off the video game series, so let's look at the basis of the TV character.
Throughout the series, there is no particular evidence to suggest that one colour is rarer than the others. In games that feature other Yoshi characters (note that the green Yoshi is, himself, a main character), there is no particular preference of colour. In fact, each Yoshi is commonly a separate colour.
The Super Mario wiki actually lists green as the most common Yoshi, but there is no source, and there is a far more reasonable explanation for seeing the green Yoshi more commonly; the green Yoshi is a main character - the other Yoshis are not. You would easily find more images of the green Yoshi through image searches, simply because hits would return concerning the main character, rather than the additional supporting characters.
The other Yoshi's first appeared in Super Mario World for the SNES. Until you reached either the special star worlds, the first time you get a red/yellow/blue Yoshi isn't until much later in the game, with the red/fire coming first, then blue/flying, then yellow/stomping. You could only use the powers of the other Yoshi's as a normal green Yoshi by eating a colored shell So yes, the other Yoshi's were RARE to find in comparison, at least back in the 90s before Yoshi Island came out.
In the game Super Mario World, you never actually needed a non-green Yoshi, though getting one meant that all turtle shells you eat will grant you the same power of your colored Yoshi. Blue eats a green or yellow shell, gets flying skill. Green eats green-only spits out a green shell.
This only happens because of Yoshi being the most famous Yoshi and being green. You only saw most of the Yoshis as green because if you search, it will bring up lots of images of Yoshi (I tested), but the colors seem to have the same level of rarity because almost every time we see other Yoshis, each one is a different color.