Edit: added more information on why composites aren't common yet.
Most of a rocket's structure consists of LOX and fuel tanks. Historically, carbon composites were viewed as too flammable to be used safely for tanks. Carbon composites failed standard tests used by e.g. NASA to determine flammability. In 2001, a study was done to re-examine this decision.
To be considered LO2 compatible, materials must be selected that will resist any type of detrimental, combustible reaction when exposed to usage environments. This is traditionally evaluated using a standard set of tests. However, materials that do not pass the standard tests can be shown to be safe for a particular application. This paper documents the approach and results of a joint NASA/Lockheed Martin program to select and verify LO2 compatible composite materials for liquid oxygen fuel tanks.
(the program was VentureStar, by the way)
And there are other issues: at LOX temperatures, carbon composites can do strange things, including igniting spontaneously when something impacts the tank. Preventing that was a matter of finding the right materials:
Tom DeLay, a researcher in the area of nonmetallic materials and processes at NASA's Marshall Space Center (Huntsville, Ala.), notes that the many composite cryogenic tank development programs initiated over the years have certainly resulted in technology improvements, yet most had budgetary and schedule constraints that did not permit researchers to identify or qualify the optimum materials for cryogenic environments.
Lockheed Martin Space Systems - Michoud Operations (New Orleans, La.) has worked for more than 20 years on various NASA programs to adapt composites for cryogenic applications.
Another roadblock in the use of composites is the need for giant autoclaves if you want to build large structures out of a composite (e.g. the SLS tanks).
"Manufacturing, especially for composites, is limited by available facility size, and the more complicated the design, the greater the cost and difficulty of manufacturing,” states the new NASA document, which was released May 11. “So concepts that are enabled by non-autoclave processing of composites and with integrated or low-cost tooling are of great importance.”
Developing a new rocket is a long and expensive process. The lack of price pressure meant companies could afford to keep using existing designs instead of having to develop new, cheaper rockets. The rocket market is far less cutthroat than the aviation market, and far smaller (so your expensive development has to be paid for by fewer sales).
With the arrival of several commercial ventures, that is changing. These days, composites are being considered by NASA, ESA and JAXA for interstage structures, cryogenic fuel tanks and other rocket parts.
SpaceX already uses carbon fiber composites for the payload fairing and interstage on the Falcon 9. Here's the interstage:
From the Falcon 9 user guide:
The Falcon vehicles’ interstage, which connects the first and second stages, is a composite structure consisting of an aluminum honeycomb core surrounded by carbon fiber face sheet plies. The interstage is fixed to the forward end of the first-stage tank.
ULA uses the same combination for their Atlas V fairing. The Atlas SRBs use a carbon-fiber casing.
This is a Falcon 9 fairing after landing:
