When I am writing a paper and listing event in order, do I use first, second, third, etc., or is first, secondly, thirdly, etc. also OK?

First, second, third - those are fine. Nothing with -ly on it, though, unless you use finally.

http://grammar.ccc.commnet.edu/grammar/transitions.htm

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Suppose you're scanning an article. Which one of the following items would stand out most for you?

When listing events in order, you can use either "first, second, third, etc." or "firstly, secondly, thirdly, etc." Both forms are commonly used and accepted in academic writing. However, there are some slight differences in usage.

If you prefer a more traditional and formal approach, using "first, second, third, etc." is recommended. This style is typically used in more formal academic and professional writing. It provides a concise and straightforward way of presenting a series of events or points.

On the other hand, if you want to add a subtle emphasis or provide a smoother transition between each point, you can use "firstly, secondly, thirdly, etc." This style allows for a more fluid and connected presentation of your ideas. It can be particularly useful when you are analyzing or discussing a series of events or actions, as it helps to establish clear relationships between the points.

Overall, the choice between "first, second, third, etc." and "firstly, secondly, thirdly, etc." depends on your personal writing style, the formality of the writing context, and the specific emphasis or coherence you want to achieve in your paper.