They are completely different.
A "conundrum" is a puzzle or riddle. I've seen magazines from the late 1800s that had a section with various kinds of puzzles, and some were called "conundrums." A dictionary says it is "a riddle in which a fanciful question is answered by a pun." So, an example would be "Why can't you ever starve on a beach?" The answer would be "because of the sand, which is there." ("Because of the sandwiches there.")
A "predicament" is a bad situation with no obvious way to get out of it. For example, "I've painted myself into a corner. I'm in a predicament. Either I have to stay here until the paint dries, or ruin the paint job by walking across the wet paint."
"Shenanigans," usually plural in the United States, means dishonest or fraudulent acts. "The building inspector allowed the apartment to be built four feet higher than the zoning law allows, and then a few months later he was driving a new Tesla. I suspect shenanigans."