Grant certainly shied away from public gatherings when possible. It does seem that the only motivation for Grant to go to dances was to please his wife and when there he typically played the role of benchwarmer.
From The Generals Wife by Ishbel Ross:
"[Julia] danced as expertly as she rode, but she soon discovered that although Ulysses presented himself regularly as her escort, she had to dance with other partners. Neither then nor later did he walk out on a dance floor without embarrassment."
From Ulysses S. Grant: His Life and Character by Hamlin Garland regarding Grant's first posting at Sacketts Harbor, NY:
"'Lieutenant Grant... and Mrs. Grant used to go to little dancing-parties, but I don t think he ever danced.'"
From The General's Wife regarding Grant's time in Detroit:
"[Julia's] dancing delighted the young officers... she never missed the weekly assemblies at the Exchange Hotel, although her husband merely looked on as she danced. Friend Palmer observed that 'he used to stand around or hold down a seat all the evening.'"
Apparently, sometimes Grant played bouncer at dances (from Garland):
"'A fellow came to a dance [in Detroit], one night, in his shirt sleeves, and set about being noisy and vulgar. Grant asked him what he meant by it. He started to make back talk. Grant told him to be quiet, and when he refused, Grant kicked him out of the door and clear out to the gate. He was a little giant physically, and a man of no words all action.'"
While stationed on the West Coast Grant even dreamed about his wife at a dance (casually ignoring him):
"I dreamed of you and our little boys the other night the first time for a long time. I thought you were at a party when I arrived and before paying any attention to my arrival you said you must go, you were engaged for that dance."
During the war Grant wrote to Julia about the children's education taking a subtle jab at dancing:
"I want the children to prosecute their studies, and especially in languages. Speaking languages is a much greater accomplishment than the little paraphernalias of society such as music, dancing &c. I would have no objection to music being added to Nellies studies but with the boys, I would never have it occupy one day of their time, or thought."
On his rest tour with his family soon after the war the Grant party stopped at West Point where he was reported in the paper to have danced in the hop at the ballroom at Roe's Hotel there:
"Last evening (Wednesday) being "hop night" a large crowd assembled in the hop room expecting to see 'General Grant dance.' About nine o'clock the General and family entered the room, and the band immediately struck up 'Hail to the Chief,' thereby giving everybody present a chance to recognize in the modest officer who now entered the room-the gallant and victorious leader of our armies during the late contests in the field. Everybody knows, or should be aware, of his ability to lead armies... but everybody did not know that the General could dance. Well, now, sober sides, don't get shocked at what I am about to disclose-he can dance, and did dance on the evening of the 26th day of July [1865], at West Point. It was amusing to see the General maneuver in the intricate movements of cotillon, but he accomplished it manfully, and it must have been particularly delightful for the young ladies to be handled in the dance by the gallant hero."
As President he tried to participate in dancing while at Long Branch, NJ (from Garland):
"During the first summer [1869] the President spent some weeks at Long Branch, exposed to all the gaieties, forms, and ceremonies of fashionable society, which he bore with most patiently, even to attempting the lancers [square dance]. It was a hard situation for a plain old soldier whose lines of life had lain far from such scenes. It brought out a curious phase of his nature: it defined his limitations. 'Madam, I had rather storm a fort than attempt another dance,' he once said to his partner."
A newspaper report of the same dance:
"General Grant danced with ease and bashfulness, and is not passionately fond of the exercise, I should imagine, he makes a very accommodating President, and does what be dislikes with the same cheerfulness as what he is fond of. His wife is a graceful and charming dancer, and I saw nothing prettier of its sort than Admiral Porter, who is very large and commanding, and with a fine carriage, turning Mrs. Grant. The latter was dressed in good taste, with a diamond necklace and ornaments. General Sherman, in the same set, danced with his usual flow of spirits, keen, mischievous eye, quick leg, and evident annoyance that the music was not more glib and noisy. He had Mrs. Michler for his partner, and seemed to envy nobody, while General Michler danced with a Miss Turnbull, the pleasant inheritor of an old family name, in due time, that Prince of the Quarter Deck, Secretary Borie, came to action, and he moved under the shadow of his set like a canoe under the hulls of a fleet of frigates. Finally, he got his foot entangled in a cable or cord of some sort, and dropped down like the sails of a nautilus. The little man arose immediately, with merriment on both sides of his face, and, 'as Mr. Dickens would say, his shins fairly winked afterward."
An image of Grant dancing at the Stetson House in Long Branch 1869...
View attachment 391389
There are other unverified accounts of Grant dancing as President with
Philippine E. Von Overstolz of St. Louis,
William Stewarts'
daughter at her Washington wedding and his daughter-in-law
Ida Grant (wife of Fred). Although the Grant's traveled the world witnessing native dances and with formal dances held at receptions in the General's honor, I found no evidence of him taking part.
I think it's safe to say that General Grant is not remembered as a prolific and highly talented dancer. That being said, it seems as though on the rare occasion that he had to "face the music" he got out there, stepped up, and got it done.