One more updated pattern. Not really carbon fiber, but it’s the most popular pattern, so I’ll give you an extra choice.
Source Atle Mo
A seamless pattern from a tile made from a jpg on Pixabay. To get the tile select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
Zero CC tileable wood texture, made by me procedurally in Neo Texture Edit.
Source Sojan Janso
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
Formed from a tile based on a drawing from 'Viaggi d'un artista nell'America Meridionale', Guido Boggiani, 1895.
Source Firkin
Simple wide squares with a small indent. Fits all.
Source Petr Šulc.
The tile this is based on can be retrieved by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
From a drawing in 'Art Embroidery', M.S. Lockwood and E. Glaister, 1878.
Source Firkin
Dark and hard, just the way we like it. Embossed triangles makes a nice pattern.
Source Ivan Ginev
An attempt for cleaning up the original image in a few steps.
Source Lazur URH
Turn your site into a dragon with this great scale pattern.
Source Alex Parker
From a drawing in 'Heroes of North African Discovery', Nancy Meugens, 1894.
Source Firkin
Prismatic Triangular Seamless Pattern III With Background
Source GDJ
Seamless pattern made from a tile that can be obtained in Inkscape by selecting the rectangle and using shift+alt+i.
Source Firkin
Inspired by a drawing in 'Kulturgeschichte', Freidrich Hellwald, 1896.
Source Firkin
A pattern formed from a squared tile. The tile can be accessed in Inkscape by selecting the rectangle and using shift-alt-i.
Source Firkin
Vector version of a JPG that was uploaded to Pixabay by theasad121
Source Firkin
Sweet and subtle white plaster with hints of noise and grunge.
Source Phil Maurer
From a drawing in 'The Quiver of Love', Walter Crane, 1876
Source Firkin
A seamless textured paper for backgrounds. Colored in pale orange hues.
Source V. Hartikainen
Here's a subtle marble-like background for use on websites.
Source V. Hartikainen
From a drawing in 'In an Enchanted Island', William Mallock, 1892.
Source Firkin
Seamless pattern formed from a tile that can be extracted by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin