Logging is extremely important to me, so many of my clients were picked with that #1 in mind. I tried at least one online client, Mibbit (on their website), but the logging just flat out refused to work, so I gave up on it. I find applications more accessible than in-browser, anyway.
Windows: I used
IceChat and
HexChat, but settled on the latter for some reason that seemed evident at the time. Even HexChat had flaws, though; maybe it's because I used the Microsoft app store or something, but I remember the logs being more difficult to get to than they should've been. I'd probably have used
mIRC if I was willing to pay, but I wasn't.
Linux: So far I mostly use
Konversation, but I've tried
Pidgin too and liked it. I think
Quassel does do logs, but for various reasons I remember it being more of a hassle than it was worth. I can't speak for the console-based clients, as I never tried them. GNOME's Polari was sleek but had no features, and gets an anti-recommendation.
OS X (macOS): Textual seems to be the gold standard, but it does have a price tag. If you plan on using IRC all the time, though, I feel like it pays itself off eventually. If I still used a Mac day to day, I expect I'd have stuck with that. Colloquy for Mac has apparently been discontinued, which is just as well - it was acceptable, but old, and more importantly, its logging was bad, because the logs were basically only readable in Colloquy itself; if you loaded them in a text editor, you had to sift through a lot of formatting.
Snak was the 'old battleship' of Mac IRC clients, but it isn't supported anymore (the company released a free registration for it; you can still get that on Archive.org), and needs an older version of the OS to run. It had a Classic Maccish aesthetic with fancy HTML logs, though.
From the stock
irc.odyssey.chat options, Kiwi is the best and well-liked by almost everyone who used it, but it's hit and miss whether it will be active or not; I suspect it updates itself when Tim's not looking, and the updates break it. Most chatters, in practice, seem to stick with Mibbit. The Java and CGI clients are for legacy support more than anything else, and if you have such an old machine, chances are you'd get a far better experience by using a dedicated IRC client.
I'm clueless when it comes to mobile. I looked at different options for my iPhone but nothing looked good; I eventually figured that maybe I'm just not meant to use IRC on mobile devices. I don't use the phone that much anyway if I can help it, so it isn't that big of a deal.