Thanks for your great question. Here's what I think about the relative frequency of different joke promotions, in games and compositions. Many such compositions are helpmates or selfmates, but the rationale for promotion choice is already a bit peculiar in those, so in this topic we'll focus on games, direct mates or studies.
Regular promotions
First, a quick review of the frequency of legal promotions, as quoted in the Wikipedia page you pointed at:
Piece %
Queen 96.9
Knight 1.8
Rook 1.1
Bishop 0.2
Couple of points:
(1) A knight promotion is arguably not an under-promotion, in that the promoted piece gets immediate access to squares that the queen does not. The most common use of a knight promotion is in hot positions where a tempo is key.
(2) Rook and bishop are definite under-promotions: they may be used to turn a draw into a win by avoiding stalemating the opponent. But they might also be used (more in compositions than in real life) to turn a loss into a draw by entering a block position.
(3) I doubt that the need for a bishop promotion is as high as 0.2%. There are certainly cases where it is the answer, but I suspect that these are dwarfed by those where B under-promotion was selected for fun. I have certainly seen GM Hikaru Nakamura do this for grins against computers.
The Dummy Pawn
The dummy pawn takes the idea of under-promotion to its logical extreme. In principle, it would mainly be used in games to turn a lost position into a draw, since (unlike bishop or rook) it doesn't rely on other pieces to be blocked. But positions have also been constructed where the dummy pawn can turn a draw into a win. Here, the defensive player would have two alternative lines, one drawing against a bishop, the other to draw against a rook. For all the drama, I don't know of any tournament where the dummy pawn rule of 1862 was actually played. There are some good compositions though, but returning to the frequency comparison, dummy pawn would come in at approximately 0.0%!
Enemy unit
Moving on to promotions to enemy unit. There are numerous 19th century joke promotion problems relying on promotion to enemy officer. According to chess historian Mario Richter, one influence here was Bilguer's Handbuch des Schachspiels, which was for many generations of chess players the reference work on chess. Especially it refers to the paragraph defining the rule of promotion:
Gelangt [der Bauer] auf die letzte Reihe des Brettes, auf welcher anfangs die Offiziere des Gegners stehen, so muß der Besitzer ihn augenblicklich umwandeln und zu einem beliebigen Offizier machen. Hierbei kann man eine zweite Dame, einen dritten Springer u.s.w. wählen; nur versteht es sich, daß der Bauer nicht in einen zweiten König verwandelt werden darf.
(If the pawn reaches the last rank, which is occupied in the starting position by the enemy officers, it has to be immediately promoted to an arbitrary officer. In this way you can have a second queen, a third knight etc, but you cannot have a second king.)
So Bilguer forgot to specify that the promoted officer should have the same colour as the pawn!
There are many joke compositions involving promotion to enemy knight, bishop or rook. If one was looking for their movement capabilities, then these might be replaced by a friendly dummy pawn instead, which cannot move at all. However, an enemy unit is that it cannot be captured by another enemy unit, in particular it blocks a flight square for the king. It can also serve to prevent stalemating the opponent, by giving him a new piece to move around.
It is normally assumed that promotion to enemy rook can be followed by castling.
So in summary, I think the enemy officer is less useful than friendly dummy pawn for getting one's self stalemated, and more useful for avoiding stalemating the opponent. The relative frequencies would reflect this, approx 0.0% again.
Open Cases
There are two open cases where I know of no existing compositions:
(1) Promotion to enemy queen. How could it be in my interest to promote a pawn to enemy queen rather than enemy rook or bishop?
(2) Promotion to enemy pawn. This would require a new rule: how does a pawn on first row move? And by analogy with the rook, can it double-step?
(Promotions to king, white or black are also found in problems, but let's follow Bilguer to exclude them from scope because the field of discussion here is already broad enough, and they would require an extension to the rules concerning how checkmate works against multiple kings.)
