They're still the cutest family in Hollywood!

Star Magazine reports, however, they won't be together for long ... the tabloid claims Gwen Stefani and Gavin Rossdale are headed for divorce! Could these two be the next in a recent string of celeb splits???!!!

Well, we sure hope not and by the looks of things Saturday during this family outing, the nuclear family of four is doing well and keepin' it together. And in fact, our photographer thought he spied the beginnings of a baby bump on Gwen ... could it be?!

Gwen Stefani must be searching for inspiration for her Harajuku clothing lines!

The singer/clothing designer went on a tour of Little Tokyo in LA Saturday with her family, perhaps on the lookout for new trends to feature in her Japanese street style-inspired clothing lines for adults and children.

Hubby Gavin Rossdale toted the pair's youngest son, Zuma, when he wasn't being pushed in the stroller by mom. And Kingston plugged along, taking in the sites and some yummy lunch as well.

Stefani's clothing lines are named after the trendy youth who hang around the Harajuku train station in Tokyo. The girls are known to wear outrangeous, colorful looks, reminiscent of anime and manga characters.

LA's gentrified Little Tokyo neighborhood isn't quite so wild, but Gwen has been known to take the kids there for weekend outings and even pick up some brightly-colored home decor in the area's boutiques.

After their downtown outing, the family went to the park to let loose. Check out the game between the boys and their mom!

Fashion designer/singer Gwen Stefani debuted her new kids line Harajuku Mini for Target last month and sales are through the roof! And the superstar mom is contributing to her bottom line, buying up some of the adorable pieces for sons Kingston and Zuma.

The boys wore matching Mini plaid pants; Kingston wore a red blazer with black collar and little Zuma paired his holiday-inspired trousers with a military-style waistcoat by Stella McCartney Kids.

The trio, along with their nanny, went to Gwen's parents' house for their usual weekend dinner.

And of course Gwen was no less fashionable in her yellow and black leopard-print sweater with black mini, tights and riding boots. This is one seriously fashionable family!

Gwen took Kingston and Zuma to Underwood Family Farms Saturday and little Kingston tore up it up on this off-road tricycle!

Zuma spent most of his time in the sandbox but enjoyed a ride in the tractor train with mom.

It's amazing how Gwen goes from sold-out stadium shows to being pulled along in a cow-shaped car by a tractor. She's so down-to-earth!

This kid always has the coolest hair!

Little rebel Kingston Rossdale can pretty much get away with any hair style he wants, and totally rock it, and this latest 'do is no exception.

The 5-year-old is already known for his edgy style, and the fact that his mom is a rock star/fashion designer definitely helps.

As the product of Gwen Stefani and Gavin Rossdale, how can this kid not be cool?

When you've been all over the world like these kids have, a bunch of fake dino's in some tar does little to impress.

So it was no wonder Gwen Stefani and Gavin Rossdale's rambunctious twosome looked beyond bored, while on an outing with the nanny at the La Brea Tar Pits in LA ,on Sunday.

Kingston sported his trademark mohawk and a vintage Billy Joel tee shirt, while his lil' bro Zuma rocked a bulky blue cast, rock star shades and adorable fedora hat.

Two-year-old Zuma spent a good portion of his time taking an important call on his iPhone 4, while his five-year-old bro tried to distracted him with kisses.

But the day wasn't a total bore. Both boys fell under the spell of a performing street musician, who captured their interest so long they gave the man a sweet tip. There's no mistaking these kids' parents are in the music industry!

Gwen and Gavin are enjoying the sites and sounds of the Cannes Film Festival in France ... from a luxury yacht anchored just off the coast! There, they don't have to deal with annoying French fans asking for their autographs every second, they just get to relax and look hot and jet ski!

Seriously ... what a great idea -- SO much better than the crowded Croisette!

Check out Gavin with little Zuma on the jet ski; it's the cutest thing ever! Click through the entire gallery below to see the Stefani-Rossdale's vacation album (by X17) ...

Gwen's in town for the annual AIDS benefit hosted by amfAR at the Hotel du Cap-Eden-Roc. The L'Oreal amfAR After Party will be hosted by Gwen Stefani, Freida Pinto, Patrick Dempsey, Milla Jovovich, and Michelle Yeoh.

Partying for a cause, we like it!

Oh, and guess who'll DJ the party? Click through to find out ...

The Rossdale boys left mom and dad at home for a fun trip to the park with grandpa.

Kingston and his little brother Zuma enjoyed their quality time with Gwen's pop at Castle Park in Sherman Oaks. These little cuties are so darn stylish! Love Kingston's knee high socks.

Gavin and Gwen's little boy is certainly a chick magnet. See him putting the squeeze on a cutie at the park ... after the jump.