Fabric-ish patterns are close to my heart. French Stucco to the rescue.
Source Christopher Buecheler
It looks like a polished stone surface to me. Download it for free, as always.
Source V. Hartikainen
Alternative colour scheme. Not a pattern for fabrics, but one produced from a jpg of a stack of fabric items that was posted on Pixabay. The tile that this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
Dark wooden pattern, given the subtle treatment. based on texture from Cloaks. https://cloaks.deviantart.com
Source Atle Mo
Zero CC tileable wood texture, made by me procedurally in Neo Texture Edit.
Source Sojan Janso
Paper model of a tetrahedron. Modelo de papel de um tetraedro.
Source laobc
Plywood Web Background background image for use in web design.
Source V. Hartikainen
The unit cell for this seamless pattern can be had in Inkscape by selecting the rectangle and using shift+alt+i.
Source Firkin
Tiny circle waves, almost like the ocean.
Source Sagive
Three shades of gray makes this pattern look like a small carbon fiber surface. Great readability even for small fonts.
Source Atle Mo
Coming in at 666x666px, this is an evil big pattern, but nice and soft at the same time.
Source Atle Mo
The green fibers pattern will work very well in grayscale as well.
Source Matteo Di Capua
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
Remixed from an image that was uploaded to Pixabay by Pixeline
Source Firkin
Recreated from a pattern found in 'Az Osztrák-Magyar Monarchia irásban és képben', 1882. To get the tile this is based on select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
From a drawing in 'A Life Interest', Mrs Alexander, 1888.
Source Firkin
Horizontal and vertical lines on a light gray background.
Source Adam Anlauf
Can never have too many knitting patterns, especially as nice as this.
Source Victoria Spahn