Nineteenth century "brogues" were heavy untanned leather shoes, commonly worn by farmers in Ireland and Scotland. They were designed for working in wet environments like bogs, and often contained rows of perforations to let water run out.
We learn in Nestor that Mulligan has lent Stephen "one pair brogues," and in Proteus Stephen thinks of the "borrowed sandals" on his feet, so it seems very likely that the shoes he is wearing have such perforations. I have not been able to locate any photographs of brogues from Ireland ca. 1900, but a descendant of these rough work shoes survives today in a form of dress shoe, apparently hailing from northern England and lowland Scotland, that retains the perforations in a purely decorative capacity. (See the photo at bottom right.) Scotland also produced a shoe called the ghillie brogue which was open at the top to allow water to drain; it fastened around the ankle to keep the shoe from being pulled off in mud. Dressy forms of this shoe are still worn by bagpipers, and dancers use them in both Scotland and Ireland. (Middle photo.)
Whatever form of brogue Joyce may be remembering, the references in his novel suggest that he is thinking of traditional rough footwear, rather than the dressy city shoes that evolved from their country cousins in the 20th century. When Cyclops imagines the Citizen as an extravagantly rough "hero" from ancient Irish times, the narrative places him in brogues: "He wore a long unsleeved garment of recently flayed oxhide reaching to the knees in a loose kilt and this was bound about his middle by a girdle of plaited straw and rushes. Beneath this he wore trews of deerskin, roughly stitched with gut. His nether extremities were encased in high Balbriggan buskins dyed in lichen purple, the feet being shod with brogues of salted cowhide laced with the windpipe of the same beast." (Salting is the first, minimal stage in preparing a hide for use as leather.)
In Oxen of the Sun the same words are applied to Alec Bannon, who has recently returned from the wilds of Mullingar: "the figure of Bannon in explorer's kit of tweed shorts and salted cowhide brogues contrasted sharply with the primrose elegance and townbred manners of Malachi Roland St John Mulligan." The parodic styles of this episode make its narrations nearly as unreliable as the parodic sections in Cyclops, but it is evident that Bannon is wearing rough country walking shoes that contrast with Mulligan's urban fashions.
Twice in Circe, Bloom appears in countryfied outfits. In the first he seems to be an urban gentleman duded up for tramping through the bogs: "In an oatmeal sporting suit, a sprig of woodbine in the lapel, tony buff shirt, shepherd's plaid Saint Andrew's cross scarftie, white spats, fawn dustcoat on his arm, tawny red brogues, fieldglasses in bandolier and a grey billycock hat." In the second passage he is an itinerant western peasant, again in brogues: "In caubeen with clay pipe stuck in the band, dusty brogues, an emigrant's red handkerchief bundle in his hand, leading a black bogoak pig by a sugaun, with a smile in his eye."
Such shoes sometimes appear on the feet of Dubliners, but not ones who are well-off. Circe affords one more example: "The brothel cook, mrs keogh, wrinkled, greybearded, in a greasy bib, men's grey and green socks and brogues, floursmeared, a rollingpin stuck with raw pastry in her bare red arm and hand, appears at the door." In Lestrygonians, Bloom thinks of the fishmonger Micky Hanlon as being rich but "ignorant as a kish of brogues." A kish, Gifford notes, is "a large square basket used for measuring turf," suggesting "that if having one's brains in one's feet means stupidity, how much more stupid a basket full of empty, rough shoes."
Given all these references, it seems certain that the old brogues which Mulligan has handed down to Stephen do not allow the poet to step about the town in style.