An email to a specific supervisor is appropriate as you have an idea of what you want to research, but only one at a time. Not everyone in a group , causing multiple busy academics to start developing your ideas with you, only to find out that your already talking to their colleagues!

Emailing supervisors is fine. Be sure to include specifics: if you're interested in their research, specify what and how it relates to your interests; if you say you've visited their webpage, mention what you've seen on it! Also, address them by name.

The background you are attempting to rise above is several emails a week that begin "Dear Sir/Madam" and say things like "I viewed your webpage" "your interests are a match for mine", and have a detailed history of letter writer but include no specifics about the adressee. When I started I replied to all messages of enquiry, but after a couple went really wrong (e.g., panicking a student was going to show up at my office door without any funding and no actual position in the dept, despite all my messages repeating "no funding" and "you need to apply to the Uni"), I've learned to only reply to those that I can tell didn't just send the same message to hundreds of people. (I'm guessing that's why I got such a clingy response - I might have been the only one of hundreds that sent a reply, and they didn't actually READ any of the content of my messages...at least that's my best guess as I can't make any sense of it otherwise)

It is also worth visiting the postgraduate info page to see what the application process is like, so that you can come off as knowledgeable when that gets discussed.

That's what I did. My supervisor was recommended by a friend who's an academic in a related area, and also by another academic in my field who I'd chatted with on twitter. I made a tentative approach by email which got a very positive response, applied formally, and now I'm halfway through my first year.

Good luck.

I'd also explain in the email how you are going to fund the PhD or if you are looking for funding. I find this bit is often missing and so I then have to start checking with the student what their status is, they usually have no money and not much clue of how to obtain funding and that makes a big difference to my willingness to put time and energy into helping them through any application process. I'm happy to support funding applications if they are excellent and in with a shot, but if there's an already sourced funding path, mention it. You can always do both (apply for funding/have back up plan).