I recommend my own chess puzzle editor https://www.apronus.com/chess/puzzle/editor.php
On this long page with many links https://www.apronus.com/chess/puzzles/ you can get an idea for what kind of puzzles my editor can create.
LiChess or Chess.com have apps accessible to Android, in which you can find an analysis board and study from there. You can save the boards as well until further use.
Plus, there are a lot of features, such as finishing game with computer, analysis from engine etc.
There is the Wikipadia page "Fairy Chess Piece" with a table of popular and well known fairies. You can re-order the table by Betza notation or Parlett notation to get an ordering by move pattern. I think this is the best resource available, given your demands.
The accepted answer does currently (2021) not work with the latest Scid versions. Bug? But it works with Scid vs PC:
In Scid vs PC, open the PGN database which you want to sort (File → Open).
Open the game list, if it is not visible already (Windows → Game List).
Create an empty Scid database via File → New (make sure it is a *.si4 database, not a *.pgn ...