I was looking for PhDs in social sciences and there are few to none studentships in that field. There is currently an opening for a project I really do like but the university is not that famous..it climbed the rankings really fast in the recent years and it is number 58 now. The supervisor is one of the top in his field, is well acknowledged and has contributed a lot to the research community.
Does the university reputation matter? Should I go for a high reputation uni and self-funding? Is there a right or wrong at the end of the day or a PhD is a PhD no matter where and what?

Thank you

When you apply for a postdoc or Lecturer post, recruiters will be looking for a candidate with an impressive publication record. Don't expect to be handed an academic post simply because you were at a Russell group university. If someone from an ex-poly has a good publication record over a Russell graduate, the former will no doubt get the job.

I agree with TheEngineer

Thanks a lot your answer was invaluable. Should I ask potential supervisors on whether I will be given an opportunity to publish reports or is it something that I do by myself? If so, how do I get such opportunities? Apologies on my naive question, I am a newbie

No need to ask your potential supervisor just yet. Most supervisors actually push their students to publish and they're expected to provide guidance and support. You will publish papers from your PhD research findings. Once your start your PhD, it's advisable to interact with senior PhD students and postdocs/research associates in your research group. You will learn a lot from them.

I would go for the funded place, if you can get it. As well as it being hugely useful financially to have the regular stipend and not have to worry too much about money while you're studying, it will look good that you managed to win a competitive studentship.

After that, I would place a good supervisor above a high-ranked university. I went to a 'prestigious' university but had an awful supervisor who was no help to me at all, and I've really struggled. If you can, check whether they have a good record of students passing their PhDs and co-authoring papers (their research group website should have details of past and present members, and a list of publications).

Thanks a lot to all of you ! Your advice is gold :)

I recommend asking as many questions as possible. Particularly if this is important to you. It is easy and understandable to be wary of "rocking the boat" at the interview stage but this can cause serious problems down the line.

For example, I wanted to know that I would be writing my own papers, able to work flexible hours and be largely left to get on with things without being asked for progress meetings every week. These were and remain red line issues for me so it was vital that my supervisor was happy on those fronts. For example, had I found out that the supervisor wrote my reports I would have terminated the PhD and started again elsewhere.

Thanks a lot for your advice ! A PhD still seems intimidating for me ( still in the applications process) so I have so many questions to ask, I am afraid that people will laugh at me : ) And wow I did not know that supervisors would write reports under their name. What would it be acceptable? Co-publishing right?

Yep this happens - sometimes they make themselves first author and the PhD student is a middle author.

And sometimes the PhD student is the the first author.

PhD student should be the first author and supervisor usually goes last.