(context: undergrad/integrated masters student in a math-related field, planning on a PhD in a CS-related field)
In Asia (at least in India and Singapore), students are generally expected to personally know (and ideally have worked with) their to-be supervisors before starting their PhD.
So it's common for undergrads to try to contact professors for summer projects -- and professors generally like it, because they get free work done and get a good feel of who may be a good graduate student for them to take later.
In the end of my second year at a UK university, I emailed a couple of professors (in other UK universities) asking for a summer project with some semi-generic topic ideas, but quickly realized this was a major faux-pas since:
- I didn't receive a reply, positive or negative, suggesting that it was regarded as spam.
- I found that some other professors -- though not the ones I had emailed -- had put notices on their website expressing their dislike of summer internship requests.
- I saw a question here that seemed to suggest that such requests are considered spam, although that may only be for generic formats for mass-emailing professors (which is not what I did).
So I realize that I don't really know how "getting into a PhD programme" works in the UK. Are you not expected to know a supervisor beforehand? Or do students generally look for other (perhaps less time-consuming for professors) ways to network with potential supervisors, like emailing them about their work?