I'm not into crosswords as I'm not great at word games and often literally don't know the word, but I enjoy simple logic puzzles, both verbal and number based. I played Sudoku quite a lot as a kid, I like nonograms/picross and one of my favourite recent puzzle game is Hexcells Infinite, which is basically a mix of nonograms and Minesweeper (without randomness) in a hex cell grid. It hit just the right balance of requiring thought and being satisfying to progress in, but still not so hard that I wasn't able to see the next step eventually (unlike medium difficulty Sudoku puzzles where I lack the foresight to keep all the different possibilities in mind) so I managed to beat the whole game without clues (which is rare for me).
Speaking of which, I really like Minesweeper. It's one of two games on my phone and I managed to get sub 100 seconds on expert, which I'm pretty happy with though it's still amateur time. I like to do a 50x50 grid with 500 bombs, which is expert difficulty but with the largest possible grid in standard Minesweeper. I've only beaten it once or twice as you need to get lucky, but it's a great timewaster when on a plane or public transport as it takes over half an hour to finish.
@Misery can also vouch for Demon Crawl, which is an awesome minesweeper/Roguelite combo. I got really sucked into it for a while and would probably still be playing if it didn't keep crashing when I sequence break it (do very well).