I took the liberty of using Dmitry’s pattern and made a version without perforation.
Source Atle Mo
One more from Badhon, sharp horizontal lines making an embossed paper feeling.
Source Badhon Ebrahim
A seamless pattern formed from a square tile. The tile can be retrieved by selecting the rectangle in Inkscape and using shift-alt-I. A version of the original with random colors.
Source Firkin
Prismatic Groovy Concentric Background 2
Source GDJ
From a drawing in 'Uit de geschiedenis der Heilige Stede te Amsterdam', Yohannes Sterck, 1898.
Source Firkin
Seamless pattern the tile for which can be had by using shift-alt-I on the selected rectangle in Inkscape.
Source Firkin
To get the tile this is based on select the rectangle in Inkscape and use shift-alt-i
Source Firkin
A light brushed aluminum pattern for your pleasure.
Source Tim Ward
Light gray grunge wall with a nice texture overlay.
Source Adam Anlauf
Remixed from a drawing in 'Maidenhood; or, the Verge of the Stream', Laura Jewry, 1876.
Source Firkin
To get the tile this is formed from select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Prismatic Groovy Concentric Background 5 No Black
Source GDJ
Remixed from a drawing in 'An Index to Deering's Nottinghamia Vetus et Nova', Rupert Chicken, 1899. The unit tile can be had by selecting the rectangle in Inkscape and using shift-alt-i
Source Firkin
Prismatic Abstract Geometric Background 4
Source GDJ
Prismatic Curved Diamond Pattern 3 No Background
Source GDJ
Heavy depth and shadows here, but might work well on some mobile apps.
Source Damian Rivas
A seamless pattern the starting point for which was a 'rainbow twist' texture in Paint.net.
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
Could be paper, could be a Polaroid frame – up to you!
Source Chaos
It’s okay to be square! A nice light gray pattern with random squares.
Source Waseem Dahman
From a drawing in 'Art Embroidery', M.S. Lockwood and E. Glaister, 1878.
Source Firkin
CC0 and a seamless pattern from a tile drawn in Paint.net .
Source SliverKnight
Did some testing with Repper Pro tonight, and this gray mid-tone pattern came out.
Source Atle Mo
This was formed by distorting an image of a background on Pixabay.
Source Firkin