1. Hoshi Coupe
214 E. 9th St. and 2801 Broadway
Japanese salon shapes mushroom-heads into seventies-shag-style cuts ($37 and up) that invite admiration from peers but don’t annoy parents.
2. Arms and Armor Room at the Met
1000 Fifth Ave.
The closest thing to Buckingham Palace in this city, with coats of arms hanging from the ceiling, mysterious figures in full armor, and spears and halberds and glaives galore.
3. The Science Barge Riverside Park at W. 70th St.
A floating “sustainable urban farm” fueled by solar power and other alternative energies teaches kids how to go green.
4. Carlos Oliveira Soccer Academy
This sometime pro and college coach models his hard-core soccer instruction on the big time, so kids kick butt in the spring and fall leagues.
5. El Quijote Restaurant at the Hotel Chelsea
226 W. 23rd St.
Surprising kid cred. The wait staff make a big show of helping little ones don bibs so they can dig into lobster claws.
6. Parsons the New School for Design
66 Fifth Ave., second fl.
Kids in fourth grade and up can sign up for manga-drawing, fashion design, and architecture classes. Saturday-morning sessions are taught by long-term faculty.
7. OTTO Enoteca Pizzeria
1 Fifth Ave.
Loud enough for parents to bring their kids without annoying anyone else. Little diners are swiftly served crayons and a map of Italy to keep them busy until a yummy dish of olive-oil gelato arrives.
8. Miette Culinary Studio
109 Macdougal St., Ste. 2
Young foodies learn how to whip up pescada a la Veracruzana (red snapper from Veracruz) and moustatsonias (almond cookies from Corfu).
9. Cake Shop
152 Ludlow St.
Hipster-parent nirvana with some 10,000 vinyl records (look, Ma, no iPod!) and chocolate cupcakes shipped in from Umanoff & Parsons bakery in Tribeca. The Sunday-afternoon all-ages shows feature kid acts like Modrocket and the Bones Royale.
10. The Original Chinatown Ice Cream Factory
65 Bayard St.
Show the kids there’s life beyond Maggie Moo’s with flavors like lychee, taro, and red bean. If they misbehave, show them the duck carcasses in nearby windows.
11. City Hall
131 Duane St.
Young’uns belly up to Saturday-morning cartoons at the bar and sup on entrée portions sized, and priced, according to age.
12. Federal Reserve Bank
33 Liberty St.
The homeworklike lesson on monetary flow in the U.S. pays off with a trip to the vault 80 feet below the city’s streets, where $140 billion in gold bars sit in storage.
34 Van Dyke St.
The former Liberty Heights Tap Room is CBGB for Brooklyn’s now well-established kid-rock movement, with bands like Care Bears on Fire and Magnolia playing afternoon gigs for all ages.
The Cove at Brooklyn Bridge Park
1 Main St.
Kids can go crabbing, fishing, and shrimping at this hidden-out-in-the-open spot. Eat at your own risk.
Miniature Golf at the Jewish Children’s Museum
792 Eastern Parkway
Tee off at an indoor range that mirrors the Jewish life cycle (minus the bris). Elsewhere, walk through a supersize Sabbath meal and talk back to a fridge that churns out Jewish jokes.