The phrase to trail one’s coat, and its variants, mean to go out of one’s way to start a quarrel or a fight.

The earliest occurrence that I have found of to trail one’s coat used in this figurative sense is from the account of a dinner to Robert Peel ¹ at Glasgow, published in the Caledonian Mercury (Edinburgh, Midlothian, Scotland) of Monday 16^th January 1837—here, pease (i.e., peace) is an interjection meaning silence! be quiet!:

In reference to the speeches of both clergy and laity, we have heard it stated that dulness formed their prevailing character. Now, from this position we entirely dissent, and sure we are that no one who carefully reads the printed report which flames in our columns of to-day, will deny to the speakers the compliment of ludicrous eccentricity, spiced with rich and varied comicality. Mr Emmerson Tennant laments the decline of fun and frolic in the Green Isle. A man may now trail his coat from morning till night in the dull arena of politics, not even the magic sound of pease will get up a bit of diversion. “Politics,” said the Hon. Member for Belfast, “in Ireland had ceased to be regarded as a mere manly amusement, or as a game of chance between skilful antagonists.”

¹ The British Conservative statesman Robert Peel (1788-1850) was at that time the Leader of the Opposition in Parliament.

The phrase to trail one’s coat, and its variants, refer to the Irish practice of dragging one’s coat behind one in the expectation that somebody will, intentionally or unintentionally, step on it and provide the pretext needed for a quarrel or a fight.

This practice was particularly associated with Donnybrook Fair—as illustrated by the following:

1-: From a humorous letter from Spain by an Irish correspondent enlisted in the British Auxiliary Legion, which was sent to that country during the First Carlist War—letter published in The Age (London, England) of Sunday 31^st January 1836:

By my sowl I think our Ginerals […] have rigistered a vow in heaven not to peril their lives any way. Howsomever, it’s no fault of some of ’em, for sure there’s one Tupper, who is only a Colonel commanding a rigment by the same token, has sint a challenge to any of the Dons to come out and fight him. Sure it’s jist like a man dragging his coat-tail after him at Donnybrook Fair and daring anny one to say pase, but divil a one is there amongst them would say that word. Och! he’s a nate lad that same Tupper—long life and whisky galore to him!—a riglar Irishman, born and bred in Guernsay.

2-: From Tom Burke of “Ours” (Dublin: William Curry, Jun. and Co. – London: William S. Orr and Co. – Edinburgh: Fraser and Co. – 1844), by the Irish novelist Charles Lever (1806-1872):

“Certain it is, nations, like individuals, that have a taste for fighting, usually have the good luck to find an adversary—and as your Emperor here seems to have learned the Donnybrook-fair trick of trailing his coat after him, it would be strange enough if nobody would gratify him by standing on it.”

The practice of dragging one’s coat behind one was also mentioned in the following:

1-: From The Morning Herald (London, England) of Thursday 22^nd November 1838:

ELEGANT IRISH AGITATIONISMS.

[Carefully extracted from the spirited orations now in course of delivery at divers dinnerings throughout old Ireland by that celebrated and well known agitatory Irishman, Daniel O’Connell Esq. ²; with explanatory notes by a particular friend of his.]
[…]
“Dark and dismal were the prospects of Ireland at the close of the late parliament. I then said, when I placed my feet upon Erin’s ground, that I would hurrah through the land for ould Ireland!—(Loud cheers).—Ireland has been robbed of her rights! Who says the contrary?”
Note.—Who dare tread upon that coat?—as the Munster man in St. Giles’s says when trailing his coat behind him in the dirt on a Sunday morning, and at the time, flourishes his shelalegh round his head, by way of agitating the Connaught boys into a riglar skrimmidge;—“who dare tread upon that coat?—Who says conthrary to what I say?—Who?”—Och leave Dan alone for agitating them up all to a froth!

² Daniel O’Connell (1775-1847) was an Irish nationalist leader and social reformer.

2-: From the Bristol Times, and Bath Advocate (Bristol, Bristol, England) of Saturday 16^th March 1839:

The Bristol Mercury is this week as martial in his mood as ambitious in his diction. He has not, as yet, to be sure, screwed up his courage to the sticking point and determined to “come out” against us […].
[…]
Our swaggering sound-and-fury friend is, we begin to suspect, like all great bullies, an arrant coward. He had, according to the “mode Hibernice,” been dragging his coat through the fair for some time back—the collective valour of the Seven Champions of Christendom seemed to be compressed into his composition—the king of beasts—he kept the forest in commotion with his roar, and shook all Broad-street, if not Bristol, with his blatant efforts; but we now discover that there was nothing about him of the lion but the skin, and nothing of the ass wanting but its proverbial patience. The separate qualities of bravo and poltroon are bad enough, yet, when combined, must exceed all ordinary contempt.

3-: From The Warder (Dublin, County Dublin, Ireland) of Saturday 11^th July 1840:

MR. O’CONNELL’S POPULARITY IN ENGLAND.

At an entertainment given at the opening of the Blackwall Railway, a few days ago, where nine hundred guests, &c., sat to table, Mr. O’Connell was refused a hearing and was heartily hooted. His Sancho, Mr. C. O’Dwyer ³, threw the contents of his card case amongst the crowd, like a Munster or Connaughtman dragging his coat at a fair, to see had any present the pluck to try a game of fisty-cuffs; but he was only laughed at for his pains, and treated as a blustering bragadocia.
We wonder has he any duties to perform in the Exchequer Court, from which he draws some hundreds a year? If so, he should attend to them, rather than follow the steps of the Quixotic Daniel.

³ Andrew Carew O’Dwyer (1801-1877) was an Irish barrister and follower of Daniel O’Connell. He was, in February 1837, appointed to the lucrative legal post of filacer of the Irish Court of Exchequer. (Here, the noun filacer designates a former officer of the Irish superior courts, who filed original writs, etc. and issued processes thereon.)

4 & 5-: From Ireland: Its Scenery, Character, &c. (London: How and Parsons, 1841), by Mr. & Mrs. S. C. Hall, i.e., Samuel Carter Hall (1800-1889) and his wife, Anna Maria Hall (1800-1881):

4-: John Coyne and Anty Casey have fallen in love with each other; but “a sort of faction-feud [has] existed for many years between the Coynes of Ballyduff, and the Caseys of Ballyran, of which each [is] a member”:

One clear moonlight night, Anty had met her lover at the old Tryst—a tree near to a Holy Well, under the shadow of which they had spent many hours together, talking over the various ‘nothings’ which time out of mind have made up the sum of lovers’ ‘somethings.’
“My heart misgives me, John,” she said; “not so much on account of my father, for sure it wouldn’t be possible to do anything with him—but my mother, John dear—my kind gentle mother, that I never told a lie to about any but you—that’s what’s grievin’ me and making my heart heavy; and I’m thinking, John, no blessing will be over us this way; and the last time I was with the priest, he told me as much; and that’s another thing, it has kept me from my duty* lately; and John agra, maybe it would be better we unsaid the words that——”
Her lover would not permit her to finish the sentence. “Unsaid the words!” he repeated; “do you mean, Anty Casey, that we should unsay the promise we made kneeling by that blessed well, to each other, in the sight of God, with his stars looking down upon us; haven’t we the same hearts in our breasts, the same feelings towards each other? the Coynes and the Caseys are not farther off than they were. At the very last fair-day, though hurling Casey dragged his coat through the fair-green, daring a Coyne to touch it, did I lift a finger to him? and for whose sake did I stand back, with the eyes of all my people on me, but for yours? And this is my thanks? Oh, Anty, I never thought it would come to this!” and he dashed himself passionately on the ground; while poor Anty, terrified at his vehemence, stood by trembling, not knowing how to appease his anger.
* Confession.

5-: The authors have asked “an intelligent countryman” to talk about the faction fights:

The man had, he confessed, many a time when a mere child, incited by the example of the faction to whom his parents belonged, nerved his little arms to cast heavy stones into the mêlée, not caring how or where they fell. We usen’t to mind a bit of a shindy in those times: if a boy was killed, why we said it was “his luck,” and that it couldn’t be helped; if a fellow trailed his coat over the fair green and dared any one to stand a foot on it, we enjoyed the fight that was sure to follow, and never thought or cared how it would end.