The world's only penis museum, in downtown Reykjavik, was created by a retired historian who developed an interest in "phallology" after he was given a bull "pizzle" as a child. It's surprisingly impressive, if you'll pardon the double entendre, with more than 280 penises from 93 marine and land animals: not just whales and seals, but giraffes and coyotes, Homo sapiens (including silver casts of the, er, members of Iceland's handball team), even "changelings" and "beach-murmurers" in the folklore room. There's also penis-inspired art, tanned whale penis hides, lampshades made from rams' scrotums, framed quotes by Herman Melville and a fun gift shop (penis-shaped Tic Tac, anyone?).

Open daily 10am-6pm. Entry 1500kr, see phallus.is/en

THE BOBBY FISCHER CENTRE, SELFOSS

This small, solemn museum, in a red-roofed storybook house 50 kilometres south of Reykjavik, celebrates the life and eccentricities of American chess player Bobby Fischer, including his historic defeat of Soviet chess grandmaster Boris Spassky in Reykjavik in 1972. News clippings and photos convey the gravity of this "Match of the Century", which saw the Cold War played out on black and white squares, but the museum also feels like fan club, a place to play a game of chess next to a replica of the board used by Fischer in 1972 then pay your respects - Fischer spent the last years of his life in Iceland and is buried up the road (he died in 2008).

Open daily 1pm-4pm May-September or by appointment. Entry 1000kr, see fischersetur.is

THE HERRING ERA MUSEUM, SIGLUFJORDUR

It might seem odd that Iceland's largest maritime museum is in a tiny fishing port on the country's remote north coast - until you learn that Siglufjordur was the "Klondike of the Atlantic" until 1968, the epicentre of Iceland's herring fishing industry. Three restored waterfront buildings, and a few enthusiastic local women dressed in headscarves and waterproof yellow aprons, transport you back to the days when wooden fishing boats disgorged their catch onto the dock and "herring girls" would sing as they sorted and salted fish for export in a sort of educational pantomime. The girls' cramped living quarters are a highlight, all Cary Grant posters, wireless radios and silk stockings.

Open daily 10am-6pm June-August, 1pm-5pm in May and September, and in winter by appointment. Entry 1800kr, see sild.is

ROKKSAFN: THE ICELANDIC MUSEUM OF ROCK'N'ROLL, KEFLAVIK

One of Iceland's newest museums, Rokksafn opened in 2014 in a former music venue near Iceland's international airport, making it an ideal stop on your way into or out of the country. Equipped with an iPad and the museum's own app, you can trace the history of Iceland's pop and rock scene from the 1950s when rock'n'roll was first broadcast by the nearby US military base to present-day bands and singers such as Sigur Ros, Bjork and Of Monsters and Men posing in wild Icelandic landscapes. There are also interactive displays - a drum kit, electric guitars, a karaoke booth and a mixing desk, all with headphones - to liberate your inner muso.

Open daily 11am-6pm. Entry 1500kr, see rokksafn.is/en

ELDHEIMAR VOLCANO MUSEUM, HEIMAEY

Ten years after the island of Surtsey was created by an undersea eruption in 1963, the nearby island of Heimaey, off Iceland's south coast, was almost obliterated by a volcano that began erupting early on January 23, 1973. One of Iceland's biggest natural disasters, it was also one of its best documented; while 5000 residents fled to the mainland, journalists, scientists and tourists poured in from all over the world. This new museum of remembrance built on the slopes of Eldfell, the 220-metre volcano formed by the eruption, uses incredible footage, images, artefacts and first-hand accounts as a powerful reminder of man's ongoing struggle against natural forces in Iceland.

Open daily 11am-6pm May-October and 1pm-5pm in winter. Entry 2300kr, see eldheimar.is/en/

THE NATIONAL MUSEUM OF ICELAND, REYKJAVIK

One of the oldest museums in Iceland (it first opened in 1863) uses every trick in the museum playbook, from replica rowboats and sod-roofed huts to interactive digital displays, to help you get your head around Iceland's fascinatingly complex history: from Celtic monks who first settled there in the 8th century to Viking explorers, Norwegian and Danish rulers, Iceland's conversion to Christianity in the 16th century and independence in 1944. The boutique-like gift shop is a must, too: it has everything from wooden Viking swords and quirky knitted "beard beanies" to rune jewellery and books by Halldor Laxness, Iceland's only Nobel laureate.

Open daily 10am-5pm (closed Mondays from October to April). Entry 2000kr (free with Reykjavik City Card, see visitreykjavik.is), see thjodminjasafn.is/english

See inspiredbyiceland.com

Louise Southerden travelled as a guest of Lindblad Expeditions and at her own expense.