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
Small gradient crosses inside 45-degree boxes, or bigger crosses if you will.
Source Wassim
A seamlessly tileable pink background texture.
Source V. Hartikainen
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 frame using leaves from an image that was uploaded to Pixabay by mayapujiati
Source Firkin
A criss-cross pattern similar to one I saw mown into a sports field.
Source Firkin
Formed from a tile based on a drawing from 'Viaggi d'un artista nell'America Meridionale', Guido Boggiani, 1895.
Source Firkin
The following repeating website background is colored in a blue gray color and resembles a concrete wall or something similar to it.
Source V. Hartikainen
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
Light honeycomb pattern made up of the classic hexagon shape.
Source Federica Pelzel
Derived from an image that was uploaded to Pixabay by Kaz
Source Firkin
From a drawing in 'Worsborough; its historical associations and rural attractions', Joseph Wilkinson, 1879.
Source Firkin
This one is quite simple in design, it consists of vertical stripes layered on top of a seamless texture.
Source V. Hartikainen
Adapted heavily from a JPG that was uploaded to Pixabay by Viscious-Speed.
Source Firkin
A grid of squares with green colours. Since the colours are randomly distributed it is automatically seamless.
Source Firkin
From a drawing in 'Sun Pictures of the Norfolk Broads', Ernest Suffling, 1892.
Source Firkin
Prismatic Curved Diamond Pattern 6 No Background
Source GDJ
Formed by heavily distorting part of a an image of a fish uploaded to Pixabay by GLady
Source Firkin
From a drawing in 'Hundert Jahre in Wort und Bild', S. Stefan, 1899.
Source Firkin
Prismatic Groovy Concentric Background
Source GDJ
A hint of orange color, and some crossed and embossed lines.
Source Adam Anlauf