I've read in a few places you should use first, second third rather than firstly, secondly thirdly.

I like the latter, but should I change?

Depends what it's for. I'd probably go with what you want to put. For a paper I would probably put whatever you like and if they don't like it they can change it (there are always corrections anyway!). For your thesis I would suggest asking your sup what their preference is, or do the same as I suggested for a paper (if it comes up as a minor correction it is literally a find and replace job and it's done).

If someone knows that there is a hard and fast rule on this then follow what they say, but I doubt there is an absolute rule on it.

I like the former after my sup kept on changing my writing from the latter. However, I think the former is the US standard - they just hate adverbs over there and say things like "that was real good" (I like to annoy my American girlfriend by adding "ly" everytime she doesn't use an adverb when she should).

ooh that would really annoy me Slizor - I hope you correct her dates as well (e.g. April 3rd - April THE 3rd!!!).

hmm, as my sup is not available I will be going with my version and see if my examiners care ;-)

Just checked, I put firstly, secondly etc...not sure what is actually the correct way but it's too late for me to worry now!

well I was told that I shouldn't use whilst, but while instead. So I've gone through my entire thesis and changed to 'whilst' just because I get stubborn lol. Oh well - it will be a 'find and replace' job if it comes up in the viva.

Whilst is a far nicer word that while. Fact.

Anyone else who thinks different(ly) is wrong.

One thing that (which) really annoys me is when people use which instead of that. Oh, and when people don't have a subject for "this". My favourite example (nicked, somewhat poorly, from my sup) is below.

"The states were in constant tension during the Cold War, with small conflicts bringing the threat of nuclear annihilation. This occured in 1963......"

I have to be honest, I never did get my head around when to use that or which. My sup doesn't like it when you use which without a comma in front of it, which I learned the hard way ;-)

I haven't used first, firstly, second or secondly at all in my thesis; does this mean I'm going to fail???

What I hate is when people use two 'thats' in a sentence; usually one is sufficient. I do however think that I use too many semi-colons as my two sentences in this post demonstrate ;-)

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American writing uses first, second, third...
I think British English ( do not hang me for that term, not sure what else to call it) makes more use of Firstly, secondly, thirdly than American English.
So it might depend on your audience.
Whilst is also very British English, and when used in American English tends to sound pompous in the way it does NOT when used in British English.