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
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 based on a square tile that can be retrieved in Inkscape by selecting the rectangle and using shift-alt-i.
Source Firkin
Zero CC tileable bark texture, photographed and made by me. CC0
Source Sojan Janso
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
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
This is a hot one. Small, sharp and unique.
Source GraphicsWall
Number 3 in a series of 5 beautiful patterns. Can be found in colors on the submitter’s website.
Source Janos Koos
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
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
Seamless pattern formed from a tile that can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
Used the 6th circle pattern designed by Viscious-Speed to create a print that can be used for card making or scrapbooking. Save as a PDF file for the best printing option.
Source Lovinglf
Simple wide squares with a small indent. Fits all.
Source Petr Šulc.
Three shades of gray makes this pattern look like a small carbon fiber surface. Great readability even for small fonts.
Source Atle Mo
A free green background pattern with a pattern of rhombuses on a seamless texture. Feel free to use it as a tiled background image on your web site.
Source V. Hartikainen
A chequerboard pattern with a fruit theme. The fruits are from a posting by inkscapeforum.it.
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
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
A new take on the black linen pattern. Softer this time.
Source Atle Mo
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
This seamless web background texture looks like gray stone. It's great for using as a background image on web pages, or on some of their elements. Anyway, I hope you will find use for it.
Source V. Hartikainen
Not the most creative name, but it’s a good all-purpose light background.
Source Dmitry
One week and it's Easter already. Thought I would revisit the decorated egg contest at inkscape community: http://forum.inkscapecommunity.com/index.php?topic=118.0
Source Lazur URH
One of the few full-color patterns here, but this one was just too good to pass up.
Source Alexey Usoltsev