What is the structure of “might did”?

Solution 1:

In standard English, might + verb (past) is ungrammatical. (If you think you're hearing this, it's probably "might've" you're hearing not just "might".)

For some native speakers (not me though), might did may be acceptable. Some studies have recorded a very small handful of hits for might did (and similar):

Table 4 includes five combinations of may/might/must with did, three of which are given in (10)-(12) below. This type of construction is not [a multiple modal], but it provides important information on the possible grammatical status of the initial modal in combinations. It is attested elsewhere in the literature only in Labov et al. (1968, 262), who claim that it is evidence of a type of do-support distinctive to Black Vernacular English in New York City and that the modal thus functions as an adverb. Di Paolo reports complete rejection of sentences like She still might don't like it for her 62 white speakers in Texas (nor did she observe any examples of might don't in her years of collecting data: Di Paolo, letter to the authors).

However, all examples in our corpus come from white South Carolinians, raising considerable doubt about the claim of Labov et al. that the initial modal has achieved adverbial status in Black English but not in Southern White English. Whereas our corpus from the Carolinas has five combinations of may, might, and must with did, it lacks any with do like those cited in New York City. Butters (1991, 173) notes that "must/might + do + neg. could be viewed as an early instance of linguistic divergence of [Black English Vernacular] in New York City," but, citing the occurrence of such forms in Gullah (as reported in Cunningham 1970, 84), he asserts that this is certainly open to question (see also Butters 1989, 101-03).

10. There weren't no way to be sure; could be he MAY DIDN'T want to come.
11. You MIGHT DID already do that.
12. We waited as long as we could cause we thought he MIGHT DID want to go with us.

_{The Pragmatics of Multiple Modal Variation in North and South Carolina}