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 pale olive green background with a seamless texture.
Source V. Hartikainen
Seamless pattern formed from a tile that can be had by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin
Zero CC tileable seed texture, edited by me to be seamless from a Pixabay image. CC0
Source Sojan Janso
Dark, lines, noise, tactile. You get the drift.
Source Anatoli Nicolae
Retro Circles Background 5 No Black
Source GDJ
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
Prismatic Groovy Concentric Background
Source GDJ
Remixed from a drawing in 'Maidenhood; or, the Verge of the Stream', Laura Jewry, 1876.
Source Firkin
One can never have too few rice paper patterns, so here is one more.
Source Atle Mo
Prismatic Groovy Concentric Background 4 No Black
Source GDJ
I love these crisp, tiny, super subtle patterns.
Source Badhon Ebrahim
Prismatic Abstract Geometric Background 2 No Black
Source GDJ
Prismatic Groovy Concentric Background 5
Source GDJ
I took the liberty of using Dmitry’s pattern and made a version without perforation.
Source Atle Mo
The image depicts meshed silhouettes of various things.The original image is an OCAL clipart called "Enter FOSSASIA 2016 #IoT T-shirt Design Contest" uploaded by "openclipart".Thanks.
Source Yamachem
To get the tile this is based on select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
A large pattern with funky shapes and form. An original. Sort of origami-ish.
Source Luuk van Baars
Derived from an image that was uploaded to Pixabay by Kaz
Source Firkin
Formed by heavily distorting part of a an image of a fish uploaded to Pixabay by GLady
Source Firkin
Prismatic Polka Dots Mark II 3 No Background
Source GDJ
One more in the line of patterns inspired by Japanese/Asian styles. Smooth.
Source Kim Ruddock