For many years celebrities — along with the rest of us — have traveled to the island of Anguilla.

They’ve been making a big mistake.

They should do the islands of Anguilla, for this laid-back piece of paradise offers a whole smorgasbord of out-islands.

Some are pretty quiet, some downright inaccessible, but still exploring beyond the country’s main island should be in your plans.

Consider Sombrero Island. Once the site of an active phosphate mine in the 1800s, by this century the only sign of life was a lighthouse keeper and that role has been extinguished because of automation. The island, only three-quarters of a mile long by a quarter-mile wide, was abandoned in 2001.

Dog Island, also uninhabited, features a couple of really pretty beaches. Great news here too: the whole island’s actually for sale — if you can afford the $67-million asking price.

Uninhabited Anguillita, just off Anguilla’s west end, is considered a good dive site, boasting a couple of walls and several underwater caves.

Scrub Island, off Anguilla’s other end, lays claim to at least one gorgeous beach. The island also offers great snorkeling and what many consider Anguilla’s best hikes.

For something completely different, check out Sandy Island, just off Anguilla’s north coast.

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Think green roofs and palms, alabaster sands, orange beach umbrellas, and heaven-colored seas on this cay that’s less than 300-by-80 yards in area.

But that doesn’t stop visitors (many of whom get here by a 10-minute water taxi from Sandy Ground, on a boat called, appropriately “Happiness”) from having a blast. Play beach volleyball or simply lounge. Dine on shrimp kebabs, then down a signature cocktail like the “High Tide.”

For a similar experience, though a lot farther from Anguilla itself, you might visit Prickly Pear Cays. The south side of these twin islands is stark and dramatic, while the north side of the easterly cay claims a gorgeous beach and a lunch stop called Johnno’s Prickly Pear.

Scilly Cay, a wonderful out-island retreat just offshore in the east end, boasts conch shell paths, a killer rum punch, and lobster and chicken entrees. When you’re done lunch, you can chill on the alabaster beach or maybe check out a visiting movie star, for they’ve got a helipad to host the other half.

But you’ll still feel welcome even if you’re just regular people — as long as you don’t make the same mistake that thousands have.

Next time you think about visiting the island of Anguilla, think again. Come and visit the islands of Anguilla.