Legally, a conflict of interest is a conflict between a person’s personally interests and their professional obligations. Generally, the phrase is used when it seems a person’s personal interests interfere with an impersonal matter.
The generally accepted plural is conflicts of interest; however, since the word is describing the situation and not the two interests involved it does not have a formal plural. The term is always used in the singular. Two or more persons would all have a single conflict of interest. Multiple personal conflicts would still be grouped under the single conflict of interest.
Interestingly, a century ago the term conflict of interests was the most popular form of the phrase, as shown by the ngram below.