No, most people do not get plastic/cosmetic surgery, but more compared to other cities. When I was young and going out to the private clubs, all of the girls I knew had boob jobs, and now some girls ask for them for their 16th birthday. Eek. 16th and 17th birthday nose surgery is common in some circles.
Many well-off people have minor touch up surgeries, like there is a surgeon who does an amazing little nose surgery, just a slight change, that many well-off women on the westside get. Well-off (not rich, but physicians, attorneys, etc) 35+ women, and many men tend to get a little cosmetic surgery. Like soccer mom types across LA from Pasadena to Santa Monica, a little lipo on the thighs, raise up the boobs a bit, an eye or neck lift, hair transplants are very common for professional men. It's good for their business, they look younger, and rested, and more confident. Many women I know have a little something something done at 40, especially botox as a start. I know quite a few LA cosmetic surgeons and they do very very well.
Unfortunately, some women, especially those who had aspirations as actresses, end up doing too much and look awful. They look more plastic than Barbie and it's rather disconcerting to see a 40 yr old dressed like a 16 yr old, dragging along their 5 yr old kid.
Reasons not to get cosmetic surgery besides it might not look good -- you may go under general anesthesia, which you are not supposed to do too often for safety, and personally, I wouldn't want to waste it on something I didn't really need, in case of accident or unexpected illness. Also, there are many bad plastic/cosmetic surgeons, especially those not willing to go through the additional education (it can be 8 more years of education after med school). (Any doctor can perform cosmetic surgery, even if they're not a surgeon.) I used to date a cosmetic surgeon and when we'd be at conferences it was rather scary listening to them speak about how their peers messed things up.
So no, most people in LA do not get plastic surgery. But in some circles, it's a really high percentage.