Google Keep, mostly. I write down words and fragments of songs - lines, rhymes, choruses - as they occur to me. When inspiration hits, I turn these into a song lyric, and then I sit down with my guitar or my keyboard and I come up with a chord progression and a melody, which I memorize through practice and repetition.
I learned this workflow from my mother, who is a singer-songwriter, and literally every songwriter in history, though most of them probably used notebooks instead of Keep, as I did until a few years ago. I've been using this workflow for does math 33 years now, since I was a teenager.
I think I've probably used rhyming dictionaries two or three times in that time. I would no more use any form of AI to write lyrics than I would build a robot to exercise for me. It would completely and totally miss the point of songwriting for me, which isn't producing "content" - it's a self-expressive art form.
If you find yourself wanting more than a pen and paper or a notes app for songwriting, I respectfully suggest - as someone who's been writing, performing and recording songs for the aforementioned 30+ years as well as growing up the child of a professional songwriter and also being myself a paid music critic and journalist for many years - you might not be understanding the assignment.
The most important tool you have as a songwriter - or any writer - is your ears. Listen to music. Listen to songwriters who are very good at it - there's this cat named Bob Dylan who's got a bit of a reputation for that, or Leonard Cohen or Joni Mitchell or Nick Cave or Paul Simon or Jason Isbell. Listen to what they do and figure out what they're doing.
And unless you're a born storyteller, you often need to have an interesting life to write interesting songs about. I suggest falling in love with lots of people and travelling and paying attention when you're doing these things, not just to how you feel but everything around you. Pay attention to people and write about them.
An example lyric:
I rang the New Year in / In a field out in the suburbs / Somewhere outside East Berlin / I watched the fireworks burn the night / And I wondered where you were / And if you were alright
See, that lyric exists because I actually spent New Year's Eve in a field outside Berlin. Go do that. Or get good at convincing people you have.
This isn't about software, but all the software on earth won't help you if you can't write a good song with a pen and paper. It's not enterprise software, it's not labor intensive, one person can do it sitting in a diner with a college ruled notebook and, indeed, many of the best songs have been written that way.
Or you can let an AI write it for you and dink around with the settings and call yourself a songwriter, but regardless of what you or the world tell you, you won't be. You'll actually be doing less work than most celebrity record producers and what comes out will be soulless product.
If that's what you want, great, I guess, but you certainly don't need anyone's help to make shite pseudoart. People are filling the world with it right now.