Since the spelling of the plural of "Master's degree" appears to vary depending on what style guide you follow, I thought I would post an answer that goes into more detail on the possible linguistic bases of using a singular or a plural form.
The behavior of irregular genitives such as men's/man's or child's/children's appears to provide some support for using the plural form in these kind of circumstances.
The "classifying genitive": the grammatical construction involved
The word "master's" in "master's degree" seems to be what is called a "classifying genitive".
The "genitive" affix -s is most commonly attached to an entire noun phrase, forming a genitive phrase that acts as the determiner of the following noun. (The function of "determiner" can be carried out by words like the, my, his, her,: in a phrase like "my sister's gloves", the genitive phrase "my sister" is in the same "slot" that would be occupied by her in "her gloves" or "the" in "the gloves".)
"Classifying genitives" are a bit different. In a classifying genitive, the -s affix is attached to a noun, not an entire noun phrase, and the resulting genitive is used attributively. An example is old wives' in the phrase "an old wives' tale": since old wives' is plural, we know that the preceding article an doesn't go with it, and so the overall structure has to be something like "an [[old wives'] tale]".
The principles for pluralizing classifying genitives don't seem to be entirely straightforward. I found a 1993 article that has the following discussion:
for many speakers the natural plural corresponding to (The police have just found) a man's hat, in a non-specific sense, is not some man's hats but rather some men's hats. If PHM [Professor P.H. Matthews of Cambridge] is right, and the latter case is not a classifying genitive at all but simply a genitive phrase (in determiner position) with indefinite reference, we still have to explain why the plural form men's is used when it is far from certain that more than one man is involved, particularly if we consider a case like some men's gloves, which might well be a single pair, and presumably belong to one man.
(Allerton, p. 559)
Allerton goes on to say that
the combination of genitive singular with plural head noun seems to be generally unacceptable. Combinations like these child's swings or some man's coffins or two woman's handbags seem to be unnatural, despite the fact that each object may belong to only one person or even that all objects may belong to the same person. Why? RCLM [Dr. R.C.L. Matthews of Freiburg-im-Breisgau] suggests that a kind of 'number agreement' may be involved.
(p. 562)
But at the end, Allerton notes that it may be harder to determine the behavior of nouns with regular plurals, since there is no difference in pronunciation between the genitive singular and plural:
In this discussion we have concentrated on (animate) nouns with irregular plurals, because only these have an overt distinction between genitive singular (e.g. woman's) and plural (e.g. women's). Regular nouns have no such distinction in pronunciation, despite the difference in the spelling (boy's/boys', lady's, ladies', etc.), and the non-genitive plural is also identical (boys, ladies). This gives rise to a further area of indeterminacy, particularly when a combination shows signs of being lexicalized as a compound (which Bouscaren and Chuquet (1992: 116-19) wish to place in a separate category of 'strictly generic genitives', but only when they have stress on the first element). Our original example quoted from Quirk et al. was girl's school occurring as a rival to girls' school, with some writers turning in desperation to girls school
(pp. 566-567)
Allerton suggests that girls' school is "the natural form", while forms like girl's school and girls school are the product of "the desperation of millions of language-users educated with little training in grammar and even less in the use of the apostrophe" (p. 567). But these comments seem a bit opinionated to me.
Master's degree does in fact (obligatorily) have stress on the first element for me, so it would seem to be of the correct form to qualify as a "generic genitive" construction. (I will try to find more information about this.)
Some corpus data
The Corpus of Contemporary American English (COCA) shows 401 total - 9 spoken = 392 tokens of "master 's degrees," in contrast to 49 - 9 = 40 tokens of "masters degrees". Unfortunately, I wasn't able to get any data about the spelling "masters' degrees".
The Google Ngram Viewer shows the following results for "masters degrees, master's degrees, masters' degrees":
Regardless of the grammar of the construction, the spelling "master’s degrees" does not appear to be uncommon compared to the alternatives in English texts.
Works cited
DJ. Allerton (1993) Problems of modern English grammar III classifying genitives: Putting the apostrophe in its place findings, English Studies, 74:6, 559-568.
