There's tons of options and advice for mounting cameras for shooting inside and outside of cars. Here's a few inexpensive options, and some advice from around the web, on smaller setups that won't invite unnecessary concern.
CNET has a nice discussion of how they shoot car videos for CNET on Cars. They tour many cameras and accessories used for shooting inside and outside of cars, from ENG on down to a Samsung instamatic with a front-facing screen.
You might need audio too, so check out Video Tutorials: Capturing Sound in Cars ~Clinton Harn at Zacuto. Here's part 1 on 3 simple techniques, microphone types & placements for car interior sound.
There's a large ecosystem of GoPro users and mount accessories, and safety tips for rigs attached to car bodies. Here's a test from Tom Guilmette, GoPro Hero Car Driving Test – Interior and Exterior Using Suction Cup Mounts:
In GoPro Hero3 Review! Mount it Anywhere!, Film Riot toured the possible uses of the most recent GoPro, reminding After Effects users that they can easily fix fisheye warping with the Optics Compensation effect. They later added Mount Your Camera Anywhere! VBAG Review.
Indie Mogul compares the cheaper older models with the newer in Camera Test: New GoPro Hero3 vs. Hero2 / Software Review: FilmConvert.
The GoPro isn't the only game in town, but rivals are struggling. The Contour camera system may have been better than the GoPro in many ways, but Contour closes Seattle HQ, shocks employees — execs mum on status of GoPro rival, then Contour co-founder: ‘We weren’t done showing the world what we could do’.
Techmoan is dedicated for reviews of inexpensive cameras in this class (see the Beginners Guide), plus fun spy cam reviews like the HD Pen Spycam Review. Mat of Techmoan has an in-depth review of the now 2 main competitors, GoPro HD Hero3 White vs Sony HDR-AS10 – The ULTIMATE Action Cam REVIEW.
Gizmodo posted a quick 4-way video, Hero 3 Black vs Contour+2 vs Sony Action Cam vs Drift Ghost, and a fuller comparison on Gizmodo, The Best Action Camera.
Techmoan also looks at The best action camera under $100. The SJ1000 budget 1080p multi-purpose waterproof action cam. Later referring to this budget cam, Mat concluded that “the mounting system a camera employs is every bit as important as the performance of the camera itself”! See also his look at The Mobius Camera – The do-it-all mount-anywhere 1080p Micro Cam.
A video from Videomaker, In Car Video Camera Techniques, had some old-timey advice.