Recently, I had to wait 7 hours in LAX overnight. I came in with another flight and neither left the secure area, nor did I check my bags out. All launges closed at about 12:30am. There are many airport employees, security guards, and overnight passengers around, it does not feel unsafe.

I slept in a waiting area near gate 40. A few other passengers were sleeping on the floor and seats (I don't know how they do it with the fixed arm rest..), and there are airport employees working on the hallways (cleaning, etc.); their machines can get a little noisy for short moments. It felt safe and the TSA did not come to kick people out. There are other places further away that become more quiet, such as the International area, but there is much less people around.
Btw, couples, single men, single women do sleep overnight. So others also think this is safe.

I found a sleeping spot on the floor against the windows, behind one of the large rounded columns that covers a little bit from light and noise, and with no people traffic right there.
I put documents, cell, and cash on my pant pockets, and used my backpack as a pillow; covered myself wth my jacket and got about 4 hours of nice sleep.. At 5am the shops opened again and there were lots of passengers already. I got some warm Dunkin Donught for breakfast and moved to my plane.

Btw, the satellite gates such as 52 from American Eagle, close at night. Do not go there until before your flight. There are only a few vending machines, better to get good food at the main gates.