Fiction
In Our Cousins in Ohio
(1849), one November task is cutting wool clothing scraps for rugs. The
pieces are sewn together and rolled into a large balls; when woven, the
weft may be visible, created a mottled effect. It might also be woven
in stripes/plaids or else dyed a solid color. Here, the narrator claims
that such rugs are common in America, used in the kitchens and dining
rooms of the rich, and throughout humbler houses.
"What Small Hands May Do" (1851) has a 14-year-old surprise her family with a new parlor carpet. She saves up scraps over the course of a year, cutting them into strips, and paying the local weaver $3 to make the carpet. The weaving takes four days, and the warp is included in the price. Red, blue, and white rags are specifically included to add decorative "figures." The woven carpet is pieced to fit the floor.
In the didactic story Home Comforts: Or, Economy Illustrated
(1855), a rag carpet is made up in odd moments, using scraps of old
clothes. The warp and weaving were paid for with additional rag-weft
(both weaver and the family providing the rags got half of the final
rug).
The story "Making the Best of Everything" (Peterson's, 1861) has one cousin teach another to cut old clothes for a rag carpet: the colors are cut separately and dyed, then sent off to be woven (implying visible weft). This is considered a summertime task.
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Piecing cotton and linen cabbage for the weft.
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Non-Fiction
The Harbinger (1846) newspaper describes a whole industry in New York, in which rag-pickers sort, clean, and stitch rags into 1/2" strips which are sold to weavers to make rag rugs.
Miss Leslie
(1850) disapproves of rag rugs being used in kitchens (on the ground
that they get greasy and soiled), but recommends them for other service
rooms or "where much economy is necessary." She mentions only wool
fabrics (cloth from old garments, along with baize, flannel, or scraps
of ingrain carpets).
Letters to Country Girls (1853) gives green dye instructions specifically for carpet warp (indigo over any shade of yellow), and recommends using "ley and copperas" to make a colorfast yellow rug. Mrs. Swisshelm goes on claim that the hard-packed (warp-faced) rag rugs woven by professionals wear out quickly; she recommends requesting a more "thin" and "loose" weave.
Eliza Lea's Domestic Cookery, Useful Receipts, and Hints to Young Housekeepers
(1859, 1st ed 1851) gives instructions for "How to Make a Rag Carpet."
It recommends a warp of purple, green, yellow, and red (in stripes
arranged according to your own fancy), and advises that the weft rags be
made from 2/3 wool scraps and 1/3 cotton. It also observes that carpet
warp is usually cotton, and (in name-dropping red flannel) indicates
that the appearance of the rug is affected by the color of the weft.
Agricultural prizes from Massachusetts (1853) mention a rag rug made with cotton weft submitted in the domestic manufacture division. Rag rugs (braided and woven) are mentioned in a number of similar agricultural contest reports, one of which opines:
"These [rag rugs] are articles in which the taste of the maker should combine beauty with utility to form, from unsightly and useless material, a serviceable parlor ornament. Arrangement of color should be carefully studied, and neat patterns selected or devised which are adapted to the colors at command."
The New York Institution for the Blind (1856) particularly recommends rag-carpet-weaving as an occupation for its graduates. The report mentions that every farmhouse in the area uses rag carpeting, and that one of its former pupils has already made a career of it. In his case, he both weaves commissioned carpets, and also sells carpeting he's already woven (pointing to a market for buying rugs, as well as preparing one's own rags and hiring a weaver). Some reform schools have the children cut the rags for carpet weft.
The Canadian Settler's Guide
(1857) praises making rag rugs for use in the 'parlor, staircase and
bed-room' of one's log house. It recommends saving cotton, linen, and
woolen clothing too worn for reuse, cutting the fabrics into 1/4" strips
(1/2" for fine muslins), and paying for it be woven on a 2-ply cotton
warp. It mentions cutting turns in the fabric to save time on piecing,
but claims that piecing perfectly straight strips gives a superior
product. Cleaning the rags before use, and dying the white cotton, is
also recommended. Recipes for dying the warp blue, yellow, and black are
also provided. [Of course, clothing that is still wearable should be given to the needy, not cut for rag rugs.]
Both Peterson's and Arthur's Home Magazine (November 1865) discuss knitting rags into rugs. Where the woven rugs can be treated as carpeting (sewn together to fill a room or used in smaller pieces), the knitted rugs are suggested as mats or bedside rugs. The tone of the article indicates that this craft is not widely practiced in America as of 1865, though the author advocates for it.
In a letter dated December 1, 1855, Catherine Blaine mentions having a "very pretty" rag carpet in the parlor/sitting room of their Seattle home. In a June 9th, 1856 letter to his siblings, her husband David Blaine asks about a previous letter, in which his mother apparently visited the local weaver (in Seneca Falls, New York), asking "But what can you be going to do
with a rag carpet? Is it to be used on Derinda's kitchen?"
Artifacts
I can't find many 19th century rag rugs in any of the usual online repositories, fewer still woven rag-rugs (hooked are more common, braided scarcer still).
The Met has a single rag rug, described as cotton and wool, c.1800-1850, and woven in stripes. Looking closely at the supplemental images, it appears to be warp-faced.
The Smithsonian has a piece of weft-faced rug from Maine (dated "late 19th century"), and a plaid cotton-rag rug from North Carolina (1832).
Historic New England has a fair few warp-faced striped woven rugs
which came up in my initial search, though they have since been updated
without reference to rags (and the close zoom appears to have a yarn
fill rather than rags in most cases). And then, the holy grail (shown above): a warp-faced striped rag rug, with both warp and weft described as wool (though the overall item description reads "wool, cotton").
Conclusion
Although a humble article, rag rugs are associated with neatness and comfort. The most common structure for them (in the 1840-1865 period, in the literature I consulted) is woven, though weak evidence for braided, knit, and hooked rag rugs exists. Stripes appear to be the most common pattern in surviving rag carpets, and are also well-represented in contemporary literature. While many of the written sources treat the rags (weft) as being visible in the final product, the few surviving examples appear to favor warp-faced rugs where the rags are barely visible.
A mixture of wool and cotton materials is recommended, and most mentions of warp fiber are cotton. Surviving rugs are biased towards wool, though the descriptions are not always clear which is warp and which weft.
The most universal depiction of rag-rug making in period literature is that of the thrifty girl or woman working at home, cutting otherwise unusable wool and cotton scraps into 1/2" (or occasionally 1/4") strips, sewing these into 3-yard pieces, and rolling them into large balls, which are then sent out to be professionally woven.