Note: This is an old post no longer updated.
I have not tried every single app I could find, thus the list here may not have the best free apps that run on BB over Wi-Fi, they are the ones I know so far. And again, you can trust they actually work over Wi-Fi because I do not have a data plan. Here they are:
Bing for Blackberry m.bing.com/download/ You get Search, Maps, Top News, Business Directory and Get Directions. Pretty sweet. My favorite SE is Google, sadly Google Apps won’t run over Wi-Fi. I found some people were able to make Google Apps run by using a program called WiFiX. WiFiX is an app that will fool your BB into believing it is using your data plan, tunneling the connections through your Wi-Fi. I tried it and did not like it, I don’t think it is yet ready for prime time. I’ll keep an eye on it though.
Opera Mini mini.opera.com/ This is a nice and fast web browser, much faster than the default BB browser, the only limitation I found is you still need to use the BB browser to install apps, but I can live with that.
Snaptu m.snaptu.com Snaptu gives you a lot, it is like a portal with a list of apps you can use, the ones I care about are Facebook, Twitter and Weather, but you have others like movies and games. The Facebook and Twitter app are very basic, but again, they are free and work over Wi-Fi! For Twitter there are several options, but for Facebook this is the only one I have found so far, if you know of a better one, please let me know. Warning: People with data plan should now that there is a bug in this application that causes Snaptu to use the data channel. If you do not have a data plan you should be ok. Thanks to Kalt for pointing this out.
UberTwitter ubertwitter.com Thanks to Husss and Jason I tried this one, and I was blown away! This is by far the best free twitter client I have tried for BB that works over Wi-Fi. It has all the features I wanted to have. And it is fast too. If you use the free version it will display ads, but they are really not that intrusive. I recommend this app for Twitter.
Nimbuzz get.nimbuzz.com This is an IM app that work with the following networks: Windows Live Messenger, Skype Chat, Yahoo, Facebook, Google Talk, AIM, Myspace and ICQ.
LogicMail logicmail.org/ota This is a nice e-mail client that you can configure to get your email from any email account that supports POP or IMAP. I use it to read my Gmail. The details on how to configure it for Gmail are here.