There is a lot more to Copenhagen than its diminutive statue of a mermaid on a rock. The city is a compact metropolis that combines hip, modern design with old-fashioned Danish charm. Patrolling the patchwork of cobblestone streets and red brick buildings in its medieval center, for example, are dragons, a favored motif of Danish urban architecture. The mythical beasts glower above doorways with their wings splayed, cropping up on cornices and usurping positions occupied in other classic European quarters by demure caryatids or brawny atlantes. Four dragon tails intertwine to form the spire that rises high above the old Stock Exchange built by the Renaissance King Christian IV.

This is a city where the new harbor side opera house, designed by Henning Larsen, looks like a massive square spaceship preparing for liftoff, where slick Bang & Olufsen stores appear on practically every corner, and where home design shops ooze so much sleek functionality that they leave you pining for a bit of Louis XIV frivolity. My sweep of the place left me feeling that the sources of Copenhagen’s cool go very far back indeed, however – at least as far back as the Vikings.

Armed with a Copenhagen Card, a handy transportation and attraction pass ($35 for a 24-hour pass or $70 for a 72-hour pass; buy one in Copenhagen at the airport, train stations, or tourist offices, or online at www.copenhagencard.com), I wandered into the Danish National Museum, where I was struck by the displays of horned bronze helmets (originally decorated with feathers) and Viking-era horns whose sinuous curves tapered into graceful, almost gossamer swirls. There were also jagged 10th-century rune stones with messages etched in red, and a whole room full of the hollowed-out trunks of oak trees that the Danes of yore used as coffins. If a rejection of machine aesthetics in favor of craftsmanship and thorough knowledge of materials are central to Danish design, could there be any better example of it than this (Ny Vestergade 10, 45-3313-4411, www.natmus.dk; admission about $8 or free with Copenhagen Card)?

My next stop was the Museum of Danish Resistance, where one learns about the German occupation of Denmark – which the Germans nicknamed “the whipped cream front” – and the rescue of the Danish Jews in 1943. Among the numerous displays is a letter of protest against Nazi raids on Danish Jews from local bishops that was read aloud in churches on October 3, 1943, by which time a rescue effort in which more than 7,000 Jews were evacuated to Sweden in fishing boats was well under way (Churchillparken, 45-3313-7714, www.frihedsmuseet.dk; admission $6.50 or free with Copenhagen Card).

Another must-see is Rosenborg Castle, the beautiful but understated Renaissance home of the Danish royal crowns and crown jewels. Perhaps it is the absence of forced majesty that has earned the Danish monarchy the right to call itself the world’s oldest. That said, the collections at Rosenborg, which span the reign of Christian IV to that of Frederick VII, are splendid (Oster Voldgade 4A, 45-3315-3286, www.rosenborgslot.dk; $10.50 or $6.50 with Copenhagen Card). For maximum garden splendor, however, bypass the castle gardens and head to the lush Botanical Gardens directly across the street instead.

The ancient treasures of Copenhagen are alluring, but it’s that famously spare look unique to modern Danish design that has turned the town into a trendy design mecca. The Dansk Design Center (H.C. Andersens Boulevard 27, 45-3369-3369, www.ddc.dk; $6.50 or free with Copenhagen Card), a modern exhibit space across the street from Tivoli (the urban amusement park and garden that inspired Walt Disney), offers a periscope look into the current Danish thinking on design. Stop in for a Danish open-faced sandwich at the cafe, whose decor references peasant styles with a series of pine tables. During my visit, a temporary exhibit called “Flow” in the center’s basement invited visitors to take a basket and shop for fake goods such as a tin of “presence” or a bottle of 100 minutes.

But smart contemporary design isn’t only contained in Copenhagen’s museums: Danish design permeates everything in the city, including shop windows, everyday objects, interiors, and even mass transit. The Copenhagen Metro is so smooth, modern, and immaculately clean, with gray-toned stations displaying an effortless subterranean chic, that one can’t help but feel sorry for New York’s aesthetically underserved straphangers.

The trains are similarly streamlined. It’s worth hopping aboard even if you have no particular destination in mind, although existentialism has little place in the Danish mind-set. One place worth the trip is the Louisiana Museum of Modern Art (Gammel Strandvej 13, Humlebaek, 45-4919-0719 www.louisiana.dk; $12), on the North Zealand coast about a 30-minute train ride north of Copenhagen, followed by a 10-minute walk from the Humlebaek/Louisiana Station. The museum’s permanent collection includes works by Warhol, Picasso, and Rauschenberg, but what steals the show is the park-like setting with sweeping views across the Sound of Sweden. There are two exhibitions running concurrently until September 18, one focusing on French architect Jean Nouvel (with indoor and outdoor installations) and another featuring the vibrant pop art-style paintings of New York artist Michael Bevilacqua.

Back in Copenhagen, the main shopping street is the car-free Stroget, which cuts an east-west swath across the center of the city, but it is too crowded with tourists to offer real enjoyment. A more interesting walk is through the neighboring Gronnegade quarter, which has shaped up as the trendiest shopping section in town for Scandinavian home design and fashion. Likewise, you can shun Nyhavn, the pretty but crowded cafe-lined canal that shows up in all the tourist brochures, for a more authentic scene in the Christians havn or Vesterbro districts. The latter, west of Tivoli, was once known for its red lights and butcher shops but is now home to hip bars and appealing cafe-restaurants such as Apropros (2 Halmtorvet, 45-3323-1221) and the Carlton (14 Halmtorvet, 45-3329-9090, www.carltonkbhv.dk), which offers a terrific club sandwich.

To see some of the most interesting recent design in Copenhagen, visit the Hotel Fox, which is equal parts shelter and artistic declaration (Jarmers Plads 3, 45-3395-7755, www.hotelfox.dk; rooms from $153 to $263). The 61-room hostelry on the edge of the Latin Quarter actually has its genesis in a car – the Volkswagen Fox. The automaker handed German company Event Lab the task of finding an innovative way to launch its newest small car, and Event Lab proposed overhauling an existing three-star hotel to host parties and journalists over the course of a month before putting the space back in the hands of its original Danish owners. The result is the Hotel Fox, which opened in the spring.

Twenty-one young artists, illustrators, graffiti artists, and graphic designers from all over the world descended on Copenhagen to strip the rooms and in a matter of days paint, lacquer, and otherwise etch into their own artistic signatures, which run the gamut from understated to outlandish. About the only places left for traditional Danish functionality are the very ergonomic bathrooms, which are the same regardless of room style.

Venezuelan artist Masa christened Room 115 “Imataca” and turned it into “a relaxed space where the green organic forms camouflage the urban surroundings with a rain forest ambience and living forms come alive in the dark.” Stylized rain forest creatures in various shades of green with glow-in-the-dark eyes adorn the walls, floors, ceilings, and even the bedspread. Room 107, called “Good Spirits,” is decorated by the Australian design collective Rinzen. Inside, the good spirits of sleep (boldly rendered goblins who look a bit like Casper the Friendly Ghost) “fluidly roam the walls and ceiling, oozing good fortune and freedom.”

Breakfast is served in the reception area, where there are no tables per se, just nifty little tray holders that fold up from gently reclining low-slung blue and white polyurethane sofas. The morning spread is not minimalist: There is an impressive selection of loose-leaf teas, juices, including fresh-squeezed strawberry and raspberry, yogurt with berries in transparent glass jars, and delicious freshly baked croissants and (naturally) Danishes, square-cut and served in wax paper. You pile up your little high-tech white tray, take a seat, and eat to the beat of the soft electronic music.