The name under which “SpaceX” is incorporated is “Space Exploration Technologies Corp.”, so this is the name to which patents would be assigned. A query for patents assigned to them yields 17 patents, most related to communication satellite technologies. There is a rocket engine patent, 7,503,511, “Pintle injector tip with active cooling”.
The fact is, however, when it comes to rocket propulsion and structures technology, there is very little new which has not been patented long ago and come into the public domain, or become common practice after having been used for decades. There are few new ideas in the rocket business: the central concept of the SpaceX Raptor, full-flow staged combustion, was tested in the Soviet Energomash RD-270 engine in the 1960s, but never developed into flight hardware.
Much of what is needed to build practical flight hardware tends to be not fundamental technologies but “secret sauce” (materials, manufacturing techniques, testing, etc.) which are better protected, if at all, by trade secrets instead of patents.
In addition to trade secrets – which other people are not supposed to know – there is know-how, which can only be learned by doing.
The classic example is grandmother’s favorite recipe. We can have her secret recipe, but what we produce in the kitchen is very unlikely to taste as good as it did when grandma made it – at least, not until we have made the recipe a dozen times.
The loss of know-how is one of the hidden costs of off-shoring and out-sourcing. Usually, people don’t recognize that cost until it is too late. But that is really another discussion.
An excellent example of this is the RD-180 engine which was licensed from Russian company NPO Energomash for use in the Atlas V launcher. The license agreement created a joint venture, RD AMROSS, to procure the engines, and included provision of all information to permit domestic manufacturing of the engines should supply from Russia be cut off. When relations with Russia became tense after the Crimea annexation in 2014, it was discovered that “all information” did not include the metallurgical processes and coatings needed to fabricate the components that operated in the oxygen-rich environment of the preburner and turbopumps, and former NASA administrator Michael Griffin estimated it would take five or six years and more than a billion dollars to produce flight-ready domestic versions of the RD-180.
This is more a question of know-how than design specifications. The chemical and mechanical properties of the materials are readily reverse-engineered, but how they’re made and processed is another matter entirely.