I like this thread :-) Other picture book mysteries:
Something Queer is Going On, by Elizabeth Levy. This is a whole series from the 1970s. I stumbled upon one of them in a thrift store one day and immediately loved the title :-) They word count is a bit longer than today's PBs.
Moose Tracks by Karma Wilson (rhyming) could be considered a mystery.
Stop, Thief! by Adam J.B. Lane isn't exactly a mystery, but has a heist/thief element.
In making this list, I realized how many picture books have a structure that resonates with mysteries--problem/riddle, clues, reveal--but aren't technically mysteries. For example, One Cool Friend by Toni Buzzeo or Immi's Gift by Karin Littlewood.
And other picture books, like The Grey Lady And The Strawberry Snatcher by Molly Bang or Lost and Found by Bill Harley (both of which I love), feel extremely mysterious, but don't have the traditional "trappings" of a mystery.
I think the picture books that are tough to find, and that I'm currently interested in writing, are those that have both the structure and the trappings of a mystery: a riddle or question, clues, and an opportunity for deliberate problem-solving.
. . . All in 500 words or less, right? ;-)
I look forward to reading the titles that everyone mentioned!