Correct spelling, explanation: rarely functions in modern English as an adverb, but it is a word with a long linguistic history as it probably comes from the Latin word rarus and from an Old French form rere, which had the meaning of unusual or not happening often. Other similar words to rarely are seldom or hardly ever.

Definition of rarely :
1. adverb – not often,
I love going to parties, but I rarely have time to go out with friends.

Collocations with rarely :
A few common collocations include: very rarely, rarely found, and rarely seen.
I very rarely see him as he works abroad and visits our home town only on special occasions.
The scientists are thrilled as this particular kind of species is rarely found in this region.
She is a famous actress but is rarely seen in public as her privacy is extremely important to her.

Incorrect spelling, explanation: rarely is a difficult word when it comes both to its spelling and pronunciation. The word actually combines the adjective rare with a typical adverb ending, which is -ly. Together they form an adverb rarely. However, the word is commonly misspelled as rarley, which is wrong.