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
Made by distorting a simple pattern using the 'sin waves' plugin for Paint.net and vectorising in Vector Magic
Source Firkin
A seamless pattern formed from miutopia mug remixes on a tablecloth.
Source Firkin
A seamless pattern formed from a sports car on clker.com. To get the tile, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Semi-light fabric pattern made out of random pixels in shades of gray.
Source Atle Mo
A blue gray fabric-like texture for websites. An yet another fabric-like texture. It has subtle vertical and diagonal stripes to it.
Source V. Hartikainen
Light square grid pattern, great for a “DIY projects” sort of website, maybe?
Source Rafael Almeida
Remixed from a drawing in 'Chambéry à la fin du XIVe siècle', Timoleon Chapperon, 1863.
Source Firkin
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
Zero CC tileable pine bark texture, photographed and made by me. CC0
Source Sojan Janso
Remixed from an image that was uploaded to Pixabay by Pixeline
Source Firkin
From a drawing in 'Picturesque New Guinea', J Lindt, 1887.
Source Firkin
Prismatic Groovy Concentric Background 2
Source GDJ
From a drawing in 'A Guide to the Guildhall of the City of London', John Baddeley, 1898.
Source Firkin
A seamless pattern from a tile drawn in Paint.net and vectorised in Vector Magic
Source Firkin
Design drawn in Paint.net, vectorised using Vector Magic and finished in Inkscape.
Source Firkin
A seamless pattern drawn originally in Paint.net by distorting a slice of background pattern 116 and copying the resulting triangle numerous times.
Source Firkin
Bright Multicolored Floral Background by Karen Arnold from PDP.
Source GDJ
A dark striped seamless pattern suitable for use as a background on websites.
Source V. Hartikainen
A nice and simple white rotated tile pattern.
Source Another One
Derived from a PNG that was uploaded to Pixabay by nutkitten
Source Firkin
Not the most creative name, but it’s a good all-purpose light background.
Source Dmitry
Black version of a pattern that came out of playing with the 'light rays' plug-in for Paint.net
Source Firkin
A grid of squares with green colours. Since the colours are randomly distributed it is automatically seamless.
Source Firkin
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso