Buy in the UK, it is cheaper. Columbus Direct (http://www.columbusdirect.com/) are pretty good. I used to use Lloyds (http://www.lloyds.com/) but they have become expensive. I have done a lot of international travel for work and travel insurance is one of the expenses that have never been reimbursed, so I have always been thorough but tried not to pay top premiums because there is no actual benefit. However, do not buy a cheap cover because if something goes wrong you'll get clause this and that thrown at you and recover only a proportion of lost luggage, hospitalisation and so on.
Apart from that, check on your Hepatitis A and B and other vaccinations you'll need (depends where you are going but there are plenty of advisory websites that tell you waht for where) and get the jabs moving well before you go. If you do not have all required jabs, even if you get something else, the insurance companies would do you.
Short stays in most places are fine, Singapore makes Europe look like a filthy dump in fact, and low risk for just about everything - including or especially driving. Mind you, don't bother there, the place is small with excellent public transport. With over 40 years of trips and living in South America for longish periods I have only had a couple of little things, so pay your insurance and basically expect to write off as a necessary waste. Good trip anyway.