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Always use the following forms:
Correct: first, second, third, etc.
Incorrect: firstly, secondly, thirdly, etc.
Major usage sources (american heritage, merriam webster, colombia guide to standard english, oxford english dictionary) agree that both first and firstly are standard english usage. Many argue that the -ly is redundant and should be dropped for brevity, but that seems a week reason for dropping parts of words. Personally, I like the stronger parallelism of firstly,...secondly,...