I have interviewed for several positions at SpaceX in the past few months along the lines of propulsion design, manufacturing, and a few various test engineering position. SpaceX is very good about reaching out to you if you are qualified: expect a phone call from a recruiter within 30 minutes to a few days if you have truthfully framed your resume to represent the responsibilities, basic qualifications, and preferred skills & experience of the position to which you are applying.
The first phone call from the recruiter wants to know how your background relates to the position so know how to showcase yourself! The interview can either end after discussing your background, or with some technical questions. If you are given some technical questions, have a pen and paper ready to work out some formulaic relations. If this call goes well, they will schedule you to take to a hiring manager right then and there for the near future. If it does not go well (even if you think it did), they will not schedule you a time with the manager and it's almost certain you did not move on.
If you move on to the second round, be ready for one of the hardest phone screenings you will ever experience from any company. Honestly, for the last few positions I applied to where I got to this point, I would lose sleep the night before just because I knew how difficult they were going to be. They expect everybody they interview then to know the mechanics of a cantilever beam so they don't bother asking those type of questions. Instead, they will first have you run through your background and grill you in every aspect past your imagination. They ask theory questions related to your background that may not have been relevant at the time, but you better know them. Do not say you do not know for any question or kiss your candidacy goodbye. But do not make things up! It is 30 minutes of constant back-and-forth, high-energy. If you do well, you will be invited for an onsite interview.
The onsite interview is a 15-minute presentation of you relevant work followed by 15 minutes of Q&A. If you do well, they will do many interviews afterwards. If not, they just give you a tour of the facility.
There is one thing to keep in mind: if you feel that you are a solid, smart engineer, SpaceX will likely see that as well and will give you may opportunities to find your spot in their company. Keep applying to positions at SpaceX if you are not offered the job you just interviewed for! You will end up finding a job there somewhere. There is another glassdoor interview review below that says the same thing because it is true. I decided to go a top defense contractor instead of pursuing a position at this company anymore.