The classic 45-degree diagonal line pattern, done right.
Source Jorick van Hees
Colourful background achieved with gradient fills.
Source Firkin
Remixed from a drawing in 'Line and form', Walter Crane, 1914.
Source Firkin
Prismatic Geometric Tessellation Pattern 2 No Background
Source GDJ
Green Web Background, Seamless tile.
Source V. Hartikainen
Prismatic Rounded Squares Grid 4 No Background
Source GDJ
ZeroCC tileable mossy (lichen) stone texture, edited from pixabay. CC0
Source Sojan Janso
Abstract Tiled Background Extended 10
Source GDJ
Seamless Olive Green Web Background Image
Source V. Hartikainen
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
Clover with background for St. Patrick's Day. Add to a card with a doily, ribbon, a leprechaun or other embellishments.
Source BAJ
A seamless pattern created from a square tile. To get the tile, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Prismatic Curved Diamond Pattern 7 No Background
Source GDJ
Background Wall, Art Abstract, Block Well & CC0 texture.
Source Ractapopulous
More bright luxury. This is a bit larger than fancy deboss, and with a bit more noise.
Source Viszt Péter
Neat little photography icon pattern.
Source Hossam Elbialy
Seamless pattern the basic tile for which can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
Looks a bit like concrete with subtle specks spread around the pattern.
Source Mladjan Antic
Inspired by a drawing in 'Kulturgeschichte', Freidrich Hellwald, 1896.
Source Firkin
Crossing lines with a subtle emboss effect on a dark background.
Source Stefan Aleksić
Prismatic 3D Isometric Tessellation Pattern 6
Source GDJ
From a drawing in 'The Quiver of Love', Walter Crane, 1876
Source Firkin
The texture of this background image has some similarities with leather, and it's colored in a dark brown color. So, if you are looking for a dark brown background image for your website, this may be an option for you.
Source V. Hartikainen
To get the tile this is based on select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
This was formed by distorting an image of a background on Pixabay.
Source Firkin
A seamless pattern the starting point for which was a 'colour modulo' texture in Paint.net.
Source Firkin
A seamless pattern from a tile made from a jpg on Pixabay. To get the tile select the rectangle in Inkscape and use shift-alt-i.
Source Firkin