If you're looking at free hosts, IcyBoards is a good option. (Though they charge like 20$ to provide you with a single back up of your forum. And if this is something that you're not sure you agree with or want to deal with... Have a look at
Jcink Forum Hosting instead. They let users take back ups of their forums whenever they want and provide the .sql file for users too. )
In terms of other options...
InvisionFree is an ok host. It's kept relatively stable though no new upgrades are being added in. Though like IcyBoards you can get a copy of your database... but it will cost you. (Though unlike IcyBoards, they don't have a fixed cost... so if you get a large forum be prepared to pay upwards of 30$.)
Though if you're looking at InvisionFree... just go right to Jcink Forum Hosting (JFH). JFH has all the features of InvisionFree and then about 40 or so more on top of that. Plus JFH is actively being improved so new features get added to enhance the experience. (JFH is pretty much the closest experience you can get to self-hosting without actually being self-hosted.)
I'd recommend against proboards or zetaboards simply because they will trap you. There is no option to get a database from them.
You can try getting a free web host like 000webhosting or X10hosting... but (personally) I haven't had great luck with them.
They can be unreliable (see below for more on this), have really low resource limits (like if you use a script like mybb's installer it might eat up all your resources for that hour so you can't actually do anything until your resource limit resets) or come with extra conditions. Like for X10 hosting you had to log into your forum account/X10 account every X number of days or else they'd delete your account.
* X10 was terrible for me, I used them for a while in late 2011-early 2012 and in Feb 2012 my site was down for over half that month. Support was slow to respond and even when they responded to tell me the issue (I used my resource limit for the hour but then it just never reset like it was supposed to) was fixed, the issue still persisted for another 2 days. And then when that issue did get resolved it came right back... So naturally I gathered my data off that site and left.
000webhosting wasn't as bad... Just random downtimes of like 1-2 minutes here and there and some of my members couldn't access the site which became the largest reason for moving.