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
Heavy depth and shadows here, but might work well on some mobile apps.
Source Damian Rivas
From a drawing in 'Resa i Afrika, genom Angola, Ovampo och Damaraland', P. Moller, 1899.
Source Firkin
Seamless pattern the tile for which can be had by using shift-alt-I on the selected rectangle in Inkscape.
Source Firkin
Prismatic Geometric Pattern Background No Black
Source GDJ
An aged paper background tile with smeared and pressed text.
Source V. Hartikainen
Prismatic Geometric Tessellation Pattern 2 No Background
Source GDJ
Utilising a bird from s-light and some flowers from Almeidah. To get the unit tile, select the rectangle in Inkscape and use shift-alt-i.
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
Adapted heavily from a JPG that was uploaded to Pixabay by Viscious-Speed.
Source Firkin
A pale yellow background pattern with vertical stripes. The stripes are partially faded. I think this background image turned out pretty well, especially those faded stripes.
Source V. Hartikainen
Dark blue concrete wall with some small dust spots.
Source Atle Mo
Bumps, highlight and shadows – all good things.
Source Badhon Ebrahim
Prismatic Abstract Geometric Background 5 No Black
Source GDJ
Design drawn in Paint.net, vectorised using Vector Magic and finished in Inkscape.
Source Firkin
Washi (和紙?) is a type of paper made in Japan. Here’s the pattern for you!
Source Carolynne
Classic vertical lines, in all its subtlety.
Source Cody L
Number 2 in a series of 5 beautiful patterns. Can be found in colors on the submitter’s website.
Source Janos Koos
Prismatic Geometric Pattern Variation 2 With Background
Source GDJ