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 the starting point for which was a 'rainbow twist' texture in Paint.net.
Source Firkin
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
From a tile that can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
This background pattern contains worn out colorful stripes as a texture.
Source V. Hartikainen
Prismatic Curved Diamond Pattern 2 No Background
Source GDJ
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
Prismatic Geometric Pattern Background No Black
Source GDJ
To get the tile this is based on select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Prismatic Curved Diamond Pattern 8 No Background
Source GDJ
Remixed from a design seen in 'Burghley. The Life of William Cecil', William Charlton, 1857.
Source Firkin
Simple gray checkered lines, in light tones.
Source Radosław Rzepecki
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
This one is rather fun and playful. The 2X could be used at 1X too!
Source Welsley
From a drawing in 'Resa i Afrika, genom Angola, Ovampo och Damaraland', P. Moller, 1899.
Source Firkin
From a drawing in 'Sun Pictures of the Norfolk Broads', Ernest Suffling, 1892.
Source Firkin
The base gradient edited so now more details are rendered.
Source Lazur URH
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
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
A seamless pattern the starting point for which was a 'colour modulo' texture in Paint.net.
Source Firkin
A pale orange background pattern with glossy groove stripes.
Source V. Hartikainen
Tweed is back in style – you heard it here first. Also, the @2X version here is great!
Source Simon Leo
This seamless pattern consists of a blue grid on a yellow background.
Source V. Hartikainen