Carbon Fiber v2@2X #106
 Carbon  CC BY-SA 3.0

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

 More Textures
Prismatic Groovy Concentric Background 5 No Black@2X #468
 Dark  CC 0

Prismatic Groovy Concentric Background 5 No Black

Source GDJ

Fabric pattern (colour 4) #2399
 Fabric  CC 0

Alternative colour scheme. 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

Paper 3 #31
 Paper  CC BY-SA 3.0

Light gray paper pattern with small traces of fiber and some dust.

Source Atle Mo

Small Crosses@2X #116
 Stripes  CC BY-SA 3.0

Sharp pixel pattern looking like some sort of fabric.

Source Dmitry

Prismatic Curved Diamond Pattern 8 No Background #549
 Dark  CC 0

Prismatic Curved Diamond Pattern 8 No Background

Source GDJ

Floral Pattern Background 5 #219
 Noise  CC 0

PDP

Source GDJ

Transparent Square Tiles@2X #299
 Grid  CC BY-SA 3.0

The first pattern on here using opacity. Try it on a site with a colored background, or even using mixed colors.

Source Nathan Spady

Retro Squares Background 8 #398
 Noise  CC 0

Retro Squares Background 8

Source GDJ