Find the right verb to fill in the blank in the following sentence :
______ either of you seen my purse?

The pronoun 'either' is singular and requires a singular verb even though it seems to be referring to two things. 
But sometimes 'either' takes a plural verb when these pronouns are followed by a prepositional phrase beginning with 'of'. 
The above statement is particularly true of interrogative constructions:
 ''Have either of you two clowns read the script?'' or ''Are either of you taking this seriously?'' ​Let's see the tense of the given options:- 
Did: The auxiliary verb 'did' works with all persons, singular and plural, but the main verb appears in its base form. 
Has: It is the singular form of the verb. 
Have: It is the plural form of the verb.
 Was: It is the singular form of the verb. ​
Since 'seen' is the past participle form of the verb, thus 'have' will be used. 
Hence, the correct answer is option 3.