Correct spelling, explanation: rarely, which is similar in meaning to hardly ever or seldom, is a popular English adverb. Its origin can be found in rarus, which was Latin, and rere, which was an Old French word. The meaning of these words was similar to the contemporary meaning of rarely as they meant not occurring often or unusual.

Definition of rarely:
1. adverb – not happening often,
I wish I could change my bad habits, but I can’t quit smoking and I rarely go to the gym.

Expressions with rarely:
Expressions that often go with rarely are: rarely get the chance, rarely rains, and it rarely occurs.
I rarely get the chance to show my skills in this company because you always give me only minor tasks to do.
It rarely rains here in summer so it’s a perfect holiday destination.
He is so self-centered that it rarely occurs to him that other people have a different point of view.

Incorrect spelling, explanation: rarely is in fact a combination of the word rare with the -ly ending, which is a typical ending for adverbs. Therefore, the spelling of the word shouldn’t cause any problems, but many English users misspell rarely as rarley, which has no sense and is incorrect.