Hi, bosbjg. If you're flying into LAX that day, and have access to Terminal 7, you are entitled to use the Red Carpet Club. I recommend hanging out there if you can get access to the concourse. If you're originating in Los Angeles, you could ask the United agents for a security pass to the concourse, but that kinda negates the purpose of an airside lounge; you still have to get to TBIT and go through security.

In the Tom Bradley terminal, you are entitled to the cramped, meagerly stocked, and bathroom-less Asiana lounge, which is shared by SQ. There is a thread in the SQ Forum devoted to the dismal state of the lounges at TBIT, and SQ's delayed plans to open a refurbished lounge. (Search for "LAX Lounge" in the KrisFlyer forum.)

My advice is to stock up on books and magazines at the newsstand, then check out the lounge to satisfy your curiosity and snag whatever potables and snackables are on offer. Although the seats are comfortable, the room is small and stuffy, and can get quite crowded. You probably will not want to linger.

Good news on the Singapore side, though: On your way back, you'll have access to the nice SQ lounges in both terminals.