Running can be damaging to your feet, ankles, shins, knees, hip, back and neck. It's true, but it doesn't have to be. With an structurally integrated body and a keen attention to form (check out Chi Running for instructions on how to run without pain or injury, no matter your fitness level), running can be a cheep, easy, accessible, fun way to get get an excellent cardiovascular workout and tone your whole body.
Yoga is a phenomenal way of building strength, increasing mobility and balance, reducing stress, and in some classes, getting a cardiovascular workout as well. However, too few yoga teachers have the time and training to be able to instruct you safely. Even worse, many will coach you into things that are damaging. For example, Bikrum yoga teachers encourage locking your knees while in standing balancing poses. This is dangerous! Many power yoga teachers encourage deeper and deeper back bends, without teaching the appropriate core support to keep from damaging the spine.
It is important to learn safe body mechanics for your body before you attend most any vinyasa, flow, or hot yoga class. Learn how to keep your joints safe and how to tell the difference between healthy stretch and damage by taking an Iyengar yoga course, or finding a good beginners class led by a well-informed teacher (I recommend Ryan at Bow Street Yoga).
Cross Fit encourages attention to form, building strength and mobility in a variety of challenging exercises. The danger is in the coaching and culture of pushing past limits. The pressure to keep going past exhaustion is high. This is when most injuries happen.
Competition with self or others is a primary way that sports injuries occur. It is also what makes exercise fun and engaging for some people. I'm not saying that if you're competitive, you will be injured, or that you shouldn't engage in competitive activities if you enjoy them. I'm just recommending learning how your body's signals for exhaustion, strain, and pain are different from sensations of effort. If you practice listening to your body's early warning signals, you are much less likely to be injured.