In some of the translations of line 1537a, Beowulf grabs Grendel's mother's eaxle (shoulder) and in others he grabs her feaxe (hair). Here are some images of the original manuscript from Kevin Kiernan's Electronic Beowulf CD.
This image of the original manuscript [last word of line 1535 to end of line 1539] is from three-quarters of the way down folio 163v (from the Electronic Beowulf CD by Kevin Kiernan). The last two words on the third line are the phrase "Grendles modor;" [Grendel's mother] from line 1538b. You can click the image to see a 2070x414 pixel version (74k in size).
This is a larger image of the fourth word on the second line of the above image. It is clearly "eaxle" and not "feaxe".
The following Old English versions use eaxle without any further comment or footnotes:
Thomas Arnold (1876)
Howell D. Chickering Jr. (1977)
Elliott van Kirk Dobbie (1953)
James A. Harrison, Robert Sharp, [M. Heyne 1879] (1883)
Francis P. Magoun (1966)
John Porter (1993)
Benjamin Thorpe (1865)
C. L. Wrenn, W. F. Bolton (1953)
Bruce Mitchell, Fred C. Robinson (1998) use eaxle, and include this footnote:
1537 Geféng þá be eaxle '[Beowulf] grasped by the shoulder then'. As in l. 758, a verb in the first accented syllable of a-line alliterates instead of the following substantive, which would usually take the alliteration rather than the verb. Though not numerous, such half-lines occur throughout OE poetry and may signal special emphasis on the verb.
Fr. Klaeber (1922) uses eaxle, and includes this footnote:
1537a Rie. V. 24, Sweet L 2.22, 4 Edd., Morgan Beitr. xxxiii 117 feaxe. Cf. T.C. ss 26.
The Klaeber footnote notation is very difficult to understand, but he does include the following entries in his bibliography on page cxxix:
21. Max Rieger, Alt- und angelsächsisches Lesebuch. Giessen, 1861. [ll. 867-915, 1008-1250, 2417-2541, 2724-2820, 2845-2891.]
22. Henry Sweet, An Anglo-Saxon Reader. Oxford, 1876; 8th ed., 1908. [ll. 1251-1650.] 9th ed., revised by C. T. Onions, 1922.
Michael Alexander (1995) uses feaxe, and includes this footnote:
1537a Hair-pulling, feaxfeng, was a recognized insult.
Alfred J. Wyatt, R. W. Chambers (1914) use [f]eaxe, and include this footnote:
1537. [f]eaxe, Rieger: MS. eaxle. Rieger's emendation betters the alliteration, and has been adopted by Sweet, and by recent editors. Those who retain the reading góda in l. 758 would however be justified in quoting that line as a parallel to geféng þá be eaxle. To me feaxe appears also to give better sense: but this may be disputed. Mr. Wyatt writes: 'William Morris agreed with me that it debased Beowulf's character, turning a wrestle into an Old Bailey brawl. Hair-pulling is a hag's weapon.'
George Jack Edition (1997) uses [f]eax[e], and includes this footnote:
1537 feaxe. If the manuscript reading eaxle 'shoulder' is retained alliteration will fall on the verb Geféng but not on the noun eaxle, which is abnormal in the a-verse; emendation to feaxe restores alliterative regularity. In Anglo-Saxon law feaxfeng 'pulling the hair' is recognized as a form of insult, for which punishment is assigned. See Stanley (1976)
Then Jack includes the following entries in his bibliography on page 243:
Stanley, E. G. 1976. 'Did Beowulf Commit Feaxfeng against Grendel's Mother?', Notes and Queries 221, 339-340. Reprinted in Stanley 1987: 232-3.
Stanley, E. G. 1987. A Collection of Papers with Emphasis on Old English Literature. Toronto: Pontifical Institute of Mediaeval Studies.
W. J. Sedgefield (1913) uses (f)eaxe, and includes this footnote:
37 feaxe Ri., MS. eaxle.
as well as this endnote:
1537. Rieger's emendation feaxe is required by the metre, as the half-verse belongs to that class of A-type, one of the commonest in Beow., where the second strong stress bears the alliteration of necessity, while the first strong stress may or may not alliterate. Moreover in this class of half-verse the second strong stress is preceded by three or four words (none of which is a noun) usually unstressed, but in this type one of these bears the first strong stress. For examples cf. ll. 1492, 1496, 1497, 1501, 1506, 1518, 1521, 1535, etc. There seems to be an exception to this rule in l. 1441.
Then Sedgefield includes the following entry in his bibliography on page lii:
M. Rieger, Ingoevonen, Istoevonen, Herminonen, ZFDA xi, 177ff.