I was always taught that you should use first/secondly rather than firstly/secondly. However, upon realising that nobody else seemed to do so, I turned to Fowler’s Modern English Usage. Here’s what it says:
“The preference for first over firstly is one of those harmless pedantries in which those who like oddities because they are odd are free to indulge, provided that they abstain from censuring those who do not share the liking.”
I am not sure whether he is saying the people are odd or the oddities are odd: I suspect the latter. I hope so anyway!
Today, I decided to check in Dreyer’s English, a more recent publication. He says that you can use firstly, secondly and so on. However, if you use first, second etc it is not only correct, but marks you out as being pretty erudite. How come? Well, apparently first etc are flat adverbs, which I’d never heard of before. A flat adverb is an adverb without the “ly” at the end. They’re quite common, for example in the expression “Take it easy.”
Who knew?