Under the traditional rule, the indefinite pronoun each is always singular and takes a singular verb.
Each of the golfers wants to win the PGA. Each of us wishes for success.
The Views of Usage Panels
Some usage panels disagree. Consider this usage note on Dictionary.com:
When the pronoun [each] is followed by an of phrase containing a plural noun or pronoun, there is a tendency for the verb to be plural: Each of the candidates has (or have) spoken on the issue. Some usage guides maintain that only the singular verb is correct, but plural verbs occur frequently even in edited writing. Dictionary.reference.com/browse/each.
But on Bartleby.com we find:
The traditional rule holds that the subject of a sentence beginning with each is grammatically singular, and the verb and following pronouns must be singular. Thus you should say Each of the apartments has (not have) its (not their) own private entrance (not entrances). Bartleby.com/64/C001/025.html.
Pronouns Referring Back to “Each”
When the pronoun each is referred to by another pronoun, that other pronoun must appear in the singular. Thus:
Each of the NFL players wants to increase his bonus. We must recognize that each has identified his or her own priorities.
Again, some usage panels disagree:
It is also sometimes said that the pronoun each must always be referred to by a singular pronoun, but again actual usage does not regularly observe this stricture: Each member of our garden club had their own special interests. In the most formal speech and writing, singular verbs and pronouns occur more frequently than plural: Each member … had his own special interests. The use of plural forms, especially plural pronouns, has been increasing in the United States, partially because of the desire to avoid using he or his to refer to a female.
Sometimes, however, the pronoun each acts in apposition to a plural subject. In that case, the verb must be plural, for the grammatical subject always determines the number of the verb:
The coach and the quarterback each want to win the championship.