Shakespeare scholars use various criteria when discussing the popularity of specific Shakespeare plays. They do this mainly to increase understanding of how the reception of a specific play evolved over time, which does not only involve quantifiable criteria such as number of reprints and productions (or performances), but also non-quantifiable criteria expressed in criticism. I am not aware of a "standard way of assessing [popularity]".
For the period until the closing of the theatres in 1642, scholars look mainly at the number of reprints and the number of productions. This type of information is necessarily incomplete: some printings may not have survived and many productions must have gone unrecorded. Based on the number of reprints, Shakespeare's most popular play during this period was, perhaps surprisingly, Pericles, which saw 6 quarto reprints during the 17th century [Schabert, p. 525] in spite of the text's corruptness.
Information about productions or performances during the Jacobethan period is available from Philip Henslowe's "diary" (which does not mention Shakespeare), the Accounts of the Revels Office (for information about court performances), the Accounts of the Treasurer of the Chamber (also for information about court performances) [Halliday, p. 118] and occasional mentions in diaries and letters by other people. As mentioned above, this does not give us a complete account of the number of productions or performances of Shakespeare's plays.
For the period after the Restoration in 1660, scholars look mostly at productions (in the late 17th century, these were often "improved" versions) rather than editions. (Starting with Rowe in 1709, various individuals began publishing collected editions of Shakespeare's plays, instead of editions of individual plays, so scholars can't continue counting quarto editions.)
For example, scholars will note that,
Since R. Burbage, who was the first the play the part of Richard [III], almost all great English actors, D. Garrick (1741), J. P. Kemble (1783), C. Kean (1837), B. Holloway (1923) and L. Olivier (1944), among others, have celebrated great successes with this role. [Schabert, p. 398, my translation]
However, even counting productions, to the extent that information is available, is somewhat problematic, since many productions have been adaptations. For example, during the Restoration, Nahum Tate (1681) created an adaptation of King Lear that omitted the Fool and replaced Shakespeare's ending with a happy end in which Cordelia and Edgar got married [Schabert, p. 626]. This version remained popular until the end of the 18th century. But was it still Shakespeare?
Similarly, the Henry VI plays were neglected until 1680, when John Crowne created a two-part "updated" version adding anti-papal content and making other modifications that agreed more with Restoration tastes. The original text of the Henry VI plays was not performed again until the jubilee of 1864 [Schabert, p. 390].
Other Shakespeare plays that were adapted during the Restoration period are Measure for Measure, Macbeth, The Tempest, The Taming of the Shrew, Titus Andronicus, Timon of Athens, Troilus and Cressida, Richard II, Coriolanus, Cymbeline, Richard III, The Merchant of Venice and The Merry Wives of Windsor. Nineteenth-century producers would restore the original texts, whereas twentieth-century producers reverted to adaptation [Halliday, p 22-23].
Of course, adaptations can also be counted as a measure of popularity, but from the point of view of reception, scholars prefer to distinguish between productions of the original text, adaptations and other works that represent something new. For example, John Dryden's play All for Love (1677) "professes to imitate the divine Shakespeare" (Dryden's Preface) and displaced Shakespeare's Antony and Cleopatra on the stage until the early nineteenth century [Antony and Cleopatra, edited by Michael Neill, p. 25]; however, it stands at a much greater distance from Shakespeare's original play than other Restoration adaptations and is is usually treated as a work on its own. For this reason, a staging of All for Love would no longer count as a Shakespeare production (Restoration audiences might have disagreed), even though Shakespeare's influence on it can be regarded as evidence of Shakespeare's popularity.
Note that counting the number of productions and adaptations says nothing about their success. For example, John Dennis thought he could improve upon Shakespeare's The Merry Wives of Windsor and Coriolanus, but his adaptation of Coriolanus "was driven from the Drury Lane stage after three performances in 1719" [Halliday, p. 132]. The same applies to editions: we usually don't know how many copies were printed or how many remained on the shelves unsold.
Finally, counting productions, adaptations and editions is a bit like counting apples and oranges, since the printed format provides a totally different than a live production.
- Halliday, F. E.: A Shakespeare Companion 1564 - 1964. Penguin Books, 1964.
- Schabert, Ina (editor): Shakespeare Handbuch. Third, revised edition. Stüttgart: Kröner, 1992.
- Shakespeare, William: Antony and Cleopatra. Edited by Michael Neill. The Oxford Shakesepare. Oxford University Press, 1994. (Pages 23-67 discuss "The play in performance".)