I broke a finger on my left hand–it’s the one I use for writing, but not the dominant one. (I’m kind of weird that way–not ambidextrous at all; I just use my left hand for one specific thing.) It didn’t have much to do with dominance, in any case–I was catching a big rubber ball a friend threw at me, a two-handed task.
Let me tell you: you never realize how much you need your pinky (or at least how much it gets in the way) until it’s stuck in a splint.
I’m left handed, and it holds for me:
currently LH pinky in a splint.
Broke left arm twice, left elbow once, 8 different breakings of fingers on the left hand, leaned my left fore-arm on a soldering iron (smells like pork BBQ)
Whereas my right hand saw one sprained wrist. Oh, and I drilled a hole in my RH index-finger tip last week, but that was because my LH pinky is in a splint. And because I’m an idiot.
The middle finger on my left hand, which is not my dominat hand, has a very nasty scar that when I got it kept that hand out of comission.
Of course my right hand was holding the boxknife.
I did a header off the monkey bars when I was about seven and ended up with a greenstick fracture in my left forearm. Can’t remember if it was radius or ulna.
As far as handedness goes, my right is dominant unless I’m using chopsticks.
Strongly lefthanded. But I can sign perfectly well using my non-dominant right hand, even if I can’t write legibly at all using it. Does that count as being a little ambidextrous?
Anyway, hand and arm injuries…
Sliced bit off my left thumb with scissors when a babychild. Resulted in Spanish-American War method (?) of fixing it by sticking the severed bit back on and casting my arm from the thumb all the way to the elbow. I had to keep my arm upright for six weeks while things, uh, grew back together. At least now they’ll be able to ID me if I ever knock over a bank by looking at the perfectly circular scar on my thumbprint…
Broke left arm by falling on a raised bit of sidewalk when I was older.
Ditto. I haven’t broken a hand or wrist (just a toe, thankfully, left foot ftr) but I’ve injured a hand/wrist enough to need medical attention to stop the bleeding twice - both times my right, and I’m a lefty. Held the item I was using a boxcutter on in my right hand, slammed the glass door with my right hand… 
All of my scars are on my right (dominant) side, but the carpal tunnel problems that had me off work on disability for two months were in my left wrist. Not exactly a break or a strain, but the pain was so bad I couldn’t pick up an empty soda can with that hand.
I’ve sprained or partially fractured both of my wrists, at least 4 times each. I’ve also broken my left fibula (on my motorcycle – ALWAYS my left side on the bike). I’m very right handed.