It depends entirely on what extensions the provider uses to drive the website. Safari does not support Java, Flash, or some of the other proprietary web protocols. Heck, even Apple's iCloud (and old MobileMe) site does not work with Safari. Googles full desktop sites also choke a bit for mobile browses (which is why they have more limited mobile versions of many sites).
Basically, whenever you go to a site that is basically an app, you take your chances. As more and more sites start using HTML5 and redirecting to mobile safe versions this should become less of a problem. Of course, there are always those who just won't get with the program. Sadly, ISP's are often the worst culprits. Their email and other services are usually just tacked on, and they spend little time or effort to update them.
In the end you are usually better off just using their internet access and getting your email from another provider like Google, Yahoo, or even iCloud.
Safari, btw, doesn't use POP settings. That's the Mail app. When you access email via a web browser like Safari then server settings like POP, IMAP, and Exchange mean nothing.