For history, it's either local logging or having an always on client you can connect to (like a bouncer), but most people I knew would just switch their client on, chat for a while, and then close it. They didn't really care much about history.

It was pretty much the same thing with newer chat services like ICQ, AIM, MSN messenger, etc. (though I think they started offering offline messages in later versions).

As for mobile, my older Nokia phone had a Symbian IRC/XMPP client that I could use without any issues.