Anthropologie - this is a national chain store, but as well curated as a boutique shop. You won't be disappointed.

Blackman Cruz.

z gallerie

Ditto on the Z Gallerie - not always affordable, but they have some fun smaller things that would be easy to lug back in your suitcase!

Heath Ceramics and basically most stores on Beverly Blvd. Oh, just for fun, you should visit the Farmers Market on Third and Fairfax! There's lots to see there.

You should check out the flea markets around L.A. One of the best things about the city is how many of them there are to check out, and you can find lots of cool affordable items there. The ones at Pasadena City College and the Rose Bowl are the best.

Please tell me people are kidding about Z Gallerie and Anthropologie?! They are chain stores and can be accessed via the internet and are not representative of Los Angeles.
I would recommend going to stores along Melrose or Beverly Boulevard between La Cienega and Fairfax, also Abbot Kinney in Venice and Sunset Junction in Silverlake. All these areas are very LA in their own neighborhood-specific ways. The farmers markets throughout the city are great and if you time it right, the flea markets at either the Rose Bowl or the Melrose Trading Post are worth some time to check out - you might find a good deal on an old movie poster or film reel - very LA. Good luck!

Definitely go to Melrose Avenue. It is close to Hollywood.
I wrote about my adventures there on my blog: http://diydesignfanatic.blogspot.com/search/label/Melrose%20Avenue
There are so many different areas to shop in Los Angeles. It depends on the type of shopping you want to do. Remember that it takes 1 1/2 to 2 hours to get to wherever you are going in LA. (I lived there for many years).
Tripadvisor.com is another good place to see what people recommend.

Just a note: If you go to flea markets while in the U.S. make sure that you have cash with you, especially small bills. The Rose Bowl flea market is awesome, but like hessilou said, you have to time it right.

Does anyone know if Bountiful is still around? It was in Venice, iirc, and was lovely with all kinds of unusual stuff.

Rose Bowl in Pasadena
Maison Midi on LaBrea for Moroccan inspired housewares

concur hessilou.

hd buttercup and surrounding stores in culver city. it's on the pricey side but definitely worth the visit. it's basically furniture-land.

if you go to 3rd street promenade in santa monica, you will find several of the aforementioned stores there: anthropologie, zgallerie, as well as other chains such as pottery barn and restoration hardware.

I agree with Melrose, Fairfax, Third, Sunset Plaza (although expensive), Beverly Blvd, Abbot Kinney in Venice. I would add Montana Ave in Santa Monica, N. Robertson Blvd.( between Melrose and 3rd and Los Feliz. There are endless boutiques on each of these streets. The thing is you can park and walk for a couple of blocks and shop, then there will be a stretch of nothing so you need to get back in the car do the same thing. Its lots of driving. I would really organize your route. You could do Santa Monica and Venice together afternoon, then West Hollywood LA- Melrose, Robertson Beverly Blvd, Third, La Brea (one full day), then Silver Lake and Los Feliz on Hollywood blvd (btw-Vermont and Rodney), then a bunch of shops on N. Vermont (btw Hollywood Blvd and Los feliz Blvd) in a afternoon. if you go to Los Feliz head over to Hillhurst Ave to the Yuca's its a stand in a parking lot that make the best burrito, I miss Yuca's!!
Check out some of these furniture and accessories stores-
Jason Arnold for Modern
REHAB Vintage
Sunset Bazaar
Skank World-Beverly Blvd- My secret little place
Empiric- pricey but fun space
Mercado- Silver lake- housewares, very cute store
Yolk- in Silver Lake
Driftwood- Silver Lake
Pico Modern Furnishing- good selection of vintage, good shape and good prices
Twentieth Design- just for fun and to dream
NK Shop- a less expensive sister store of their 1st dibs
Ok shop-beautiful glassware and art objects

I could go on..

If you plan on traveling out of LA and into Orange County Out of Vogue and Otto in Fullerton are awesome.

Go to Abbot Kinney street in Venice. There is one cool non-chain shop after the next, and good restaurants too. I always take out of town guests shopping here and it always delights. All price ranges are covered, so you'll have fun browsing and buying. If you want to get an idea of what is there check out the district's site http://www.abbotkinneyonline.com/
Have fun!

all the 3rd St. boutiques in the mid-city area between fairfax and la cienega, specifically New Stone Age, Plastic, OK Gallery . . .

Abbot Kinney is my vote. Very LA vibe and unique finds.

Head East! Silverlake boulevard for Yolk, Lawson Flemming, Lake, coffee and a snack at LaMILL. Support small business that support local artist. Lots of great shops in Silverlake and Los feliz. Check out maphawk for listings.

Also in Venice is Luna Garcia, a ceramics studio. People seem to love their stuff. www.lunagarcia.com

Jonathan Adler recently opened a store on Montana in Santa Monica. Not sure if it's your style or interest.

Melrose Trading Post is a flea market / Fairfax and Melrose on Sundays, i think.

the Long Beach Flea Market is a good one too. I believe they are the 3rd Sunday of each month.

There are also some pretty cool vintage stores in North Holllywood and Burbank, along Magnolia Blvd. You might enjoy those as well. Welcome!!!!!

Wacko is great for cool kitch and some great art.

Sunset Junction in Silver Lake- a walk down that street is terrific for classic LA design inspiration.

And I'll support Melrose, too. Not a lot of design stores, but there is that terrific one near Alta Loma/Poinsettia, whose name I always forget.

I have never been to LA, but Grace of Design Sponge put together a city listing for it:

http://www.designspongeonline.com/2007/01/los-angeles-design-guide-2.html

Check out the furniture stores on La Brea, between Wilshire and Melrose. HAUS furniture has an awesome selection of modern, livable furniture. It's adjacent to Pink's.

I also recommend cruising Hancock Park on weekend afternoons for estate sales.