Interesting thread. I've really had to think about this.
South-east UK here.
Professional: Almost always in job interviews, usually initiated by the interviewer. If it's a panel interview, you'd shake hands with all of them. I've never been to an interview where this hasn't been the case. Perhaps it's different for more casual kinds of jobs but for office/professional jobs, this is the standard. Refusal to shake the interviewers hand - unless your hand was obviously in a cast or something - would probably create a very bad impression of you; you've failed basic interview etiquette 101 within the first ten seconds. Meeting new colleagues, a handshake may or may not be offered, and I'd think this is more prevalent among men (although the interview handshake is equally given to both genders). Business contacts - for example a salesman from one company meeting a purchaser from another - would usually shake hands on meeting and possibly on repeated meetings.
Business: For large purchases such as a house, car, new kitchen, etc, a handshake would probably be offered at the end of the deal. My medical consultant shook my hand the first time I went to see him as part of the introductions. Everyday business transactions like bank transactions, shopping, etc: no handshake.
Socially: Not on an every day basis - you wouldn't shake hands with your friends when you meet them. Meeting someone new for the first time you may shake their hand. Possibly a young man meeting his girlfriend's father would be offered a handshake. During organised social activities, you may shake hands with people on joining a new group for the first time - for example, if you joined an orchestra, the musical director would probably shake your hand while introducing him/herself.
There's also an element of congratulations to it, but this seems to be a specifically male thing. A man will often shake the hand of their male friend in congratulation on news of a promotion, engagement, birth of a baby, etc. This is often a two-handed handshake, or a handshake with one hand and a pat on the shoulder/back/arm with the other. I can't think what the equivalent is for women, perhaps a hug.
So yes. Handshakes are very, very common in this part of the world; so common that you don't even notice them happening, which is why I had to think so hard about this post!