Formal style in email is appropriate in a variety of situations: job-seeking, academia and business (especially when writing to someone more senior), and when you make a request of someone you don't know personally.
If the other person or organization has set the tone first, you may want to try to match it or be only a little more formal in a response; consider email a note rather than a full letter. For example, if I dash off a message to let a webmaster know about a problem, I may use "Hi," and "Thanks!" But my boss may send me a message without any salutation or signature and I'll nearly always use one in my response:
Jane Doe —
Body of email.
My Automatically Appended Signature Block (other methods of contact here)
You may find this reference helpful, by Adam Turner, HYU CTL - it's designed for non-native speakers in academia, but I'd recommend the general guidelines to anyone:
English Solutions for Graduate Research Writing: Formal Email [PDF], or the short version, a handout focusing on email style [PDF]. The first reference discusses subtleties in salutations:
Correct punctuation is important because it can show the relationship between the speakers. It is meaningful. When I email other English professors for the first time, I often start like this:
Dear Professor Taylor:
After we exchange email and I know a bit more about the person, such as that they are a similar age and status, I might change to
Dear Professor Taylor,
Finally, as we get to know each other or after we have met in person, I might change my greeting to
I'm not sure everyone will be sensitive to all levels of punctuation here, but you're unlikely to go wrong with this format.
This reference also gives some great examples of how to couch requests politely in formal email. The guidelines given seem appropriate for an American business audience, certainly. An excerpt:
Don’t demand or ask for a positive result, whether it is for a job or a journal article. Examples ...
[NO]: I am waiting for your answer soon.
[NO]: I expect that I get help from you.
[YES]: Thank you for considering my application. ...
[YES]: I look forward to your reply.
[YES]: If you have any questions or require further information, please do not hesitate to contact me at...