JoomlaHosting has set up and host sites with 10 to 15 different hosts, probably more. We have successfully installed Joomla with HostGator, Site5, GoDaddy, Rochen, Yahoo, and most of the other larger hosting companies. We have even installed joomla with several of the smaller hosts and several home grown servers.
It has been our experience that hosts who run cPanel work the best with Joomla. You will have fewer permission issues, more configuration options and better control over your site. cPanel provides a very easy to use interface for setting up and managing Mysql databases which is the first step when installing or transferring you Joomla site. In addtion to cPanel being a great platform you will be able to find more support for issue you may encounter with different extensions.
Rochen, HostGator, and Site5, they all run cPanel and provide excellent Joomla support. Quick and easy to set up Mysql, Joomla, and modify your .htaccess file.
Large well know hosting companies to avoid
We had a client that has a joomla site on GoDaddy and it does okay. Yahoo Hosting was the worst to get Joomla up and going. I would actively avoid Yahoo hosting b/c they do not give you access to your .htaccess file so any sort of SEF options becomes very difficult. I did get sh404SEF working for a client on Yahoo hosting.
Help keep your hosting company honest and see your real up and response times.
In addition to hosting I would look for a good uptime monitor. This service checks your site as often as every minute and informs you when you site goes down. The better ones will also tell you how quickly your site is responding. I like Pingdom.com the best for this service.
Hope this helps!