I don't delete any, but that's more down to laziness than anything. My phone stores ridiculous amounts of texts, more than I'll likely ever reach, so keeping them isn't an issue for me. I'd rather have them there for reference/reading back through for memory's sake than delete them for the hell of it.
Spam & Junk & other promotional messages immediately. (Automated usually and auto-spammed by my SMS app or Mr. Number)
Notifications older than 1 month from Bank(s), Telco(s), and other businesses if there's too much clutter.
Old messages from someone if I need to for some reason.
Messages that are otherwise useless
Old single-use authentication codes for online services if my inbox is cluttered.
Normal text messages
One-time use auth codes for up to a day
All payment confirmation texts from telco (Txt2Pay) less than a year old
I also keep an app running that denies most cold callers and allows me to report and add new ones to my blocklist called Mr. Number. It comes in handy because I can then deny new spammers.
Because my messages are often backed up semi-regularly, I don't typically lose any. My phone has also NEVER run out of space. I don't know how anybody does. I guess I don't get that many text messages. XD
I typically manage my inbox maybe once a month to cut out the automated messages I get that keep me notified of various things or grant me access to things.
I never delete texts but every so often (twice in the whole time I've had it), my phone decides that enough is enough and wipes my text history squeaky clean. And then I get to feel super lame while the only people in my texting history are my immediate family for a few weeks until another crest in my social life wavepool. :P
I text A LOT, so I delete all my texts every day to avoid clutter in my inbox. There were times that I've gone past the 200 mark which after 200 texts in a thread, the older texts get automatically deleted.