This list is based on feedback I've gotten from visitors to Oslo over the past 30 years. All of the top 10 sights are easy to get to by public transportation. Three of them are free, the rest are included in the Oslo Pass, which also includes public transportation.
10: The Kon-Tiki Museum
The Kon-Tiki Museum is devoted to the exploits of Thor Heyerdahl, an explorer who risked his life to prove some of his controversial theories. In 1947, to prove that pre-columbian South Americans could have traveled to Polynesia, Thor crossed the Pacific Ocean in a primitive balsa wood raft. You can see the original raft and filmed footage of the voyage that won an Academy Award (the Oscar is there as well). The rest of the museum covers two similar voyages across the Atlantic and Indian Oceans as well as Heyerdahl's groundbreaking excavations on Easter Island.
A visit here is a high priority for those who read about the Kon-Tiki as a kid. Kids today will also enjoy the museum because of the eye-popping visuals - real rafts that were sailed across the ocean, giant plaster copies of the monumental Moais and a whale shark by the entrance to a walk-through cave.
9: The National Gallery
Located in the center of town, the National Gallery has an international art collection that spans from antiquity to the 1950's. Although a fair share of international masters are represented, it's the Norwegian artists that are the draw, with Edvard Munch foremost. Here's a list of the top ten paintings that I like to show when I guide there. [UPDATE: The National Museum has now closed and will be moved to a new building opening in 2020]
8: The Norwegian Folk Museum
The big draw of this museum is the open air portion that has 160 buildings from different regions in Norway, including a stave church from Gol. Outside many of the buildings are Norwegians dressed in Bunads (folk costumes) who are available to answer questions. There are also extensive indoor exhibits about Norwegian folk culture. This museum is great to visit if Oslo is your only destination in Norway as it will give you some insight into the rest of the country.
7: Holmenkollen Ski Jump and Ski Museum
To understand Norway, you need to appreciate the special role that skiing has played in Norwegian life. A visit to Holmenkollen is a good place to start. Ski jumping has been done here since 1892 and the jump has been expanded 18 times, including a major improvement for the 1952 Olympics. The current jump was built in 2011 and you can take a funicular up to the top for a panoramic view of Oslo. An entrance to the ski jump tower also includes admission to the Ski Museum, which chronicles 4000 years of skiing in Norway.
To get there, you can take the Metro to the Holmenkollen Station and walk about 15 minutes uphill. Many bus tours include an outside visit that gives a view of the ski jump and a bird's eye view of Oslo.
6: City Hall
City halls are not usually tourist destinations, but the Oslo City Hall is an exception. Don't be fooled by the staid brick exterior. Once you enter, you'll see a grand hall richly decorated with giant murals depicting the history of Oslo and Norway. This hall is where the Nobel Peace Prize is handed out each year on December 10. After taking in all of the first floor, make sure you go up the stairs and make a circuit around all the rooms. They all have paintings with different themes. My favorite is the Per Krogh room . It took Per 10 years to paint all the walls and ceiling with a mural depicting the four seasons in Oslo and rural Norway.
The City Hall is downtown and free, but is sometimes closed to the public because of various functions.
5: The Norwegian Resistance Museum
Perched on the ramparts of the Akershus Fortress, Norway's Resistance Museum documents the resistance to the German occupation during WW II. There's no need for a guided tour, all the exhibits have amble text in English.
4: Fram Museum
Internationally, the Fram Ship is best known as the vessel that Roald Amundsen sailed to Antarctica when he became the first man to the South Pole. Norwegians know that the ship was custom-built for Fridtjof Nansen's 1893 voyage in which he sailed the Fram into the polar ice cap in order to let the ship drift close to the North Pole. Nansen left the ship and tried to ski to the North Pole. He didn't make it all the way, but made it further north than all previous attempts. The ship drifted 3 years in the polar ice cap before coming out on the western end. You can board the Fram and see all the cabins with original artifacts.
The museum also houses Amundsen's ship, the Gjoa, which he sailed as the first to navigate the Northwest Passage. There's a good film and lots of exhibits about the daring exploits of Nansen and Amundsen.
3: Vigeland Park
The Vigeland Park was created by Norway's greatest sculptor, Gustav Vigeland, and is the world's largest sculpture park made by a single artist. In 1929, Vigeland entered into a contract with the city of Oslo whereby the city provided him with a workshop and the funds to create whatever he wanted in the park. In return, the city got all the sculptures.
The centerpiece of the park, is a giant 50 foot monolith with 121 figures surrounded by 36 large granite sculptures. All in all, there are over 200 sculptures plus many iron wrought gates with intricate designs. The park is free, open 24/7 and is easy to get to by public transportation. To avoid crowds during the summer, try to go in the late afternoon or evening.
2: The Viking Ship Museum
There are only two places in the world where you can see original Viking ships - Roskilde in Denmark and the Viking Ship Museum in Oslo with it's two intact ships. The first is the Oseberg from ca. 800 and was owned by a Queen. The Gokstad ship, from around 890, was owned by a king who died in battle. An exact replica of this ship sailed across the Atlantic to Chicago in 1893.
Both of the ships were found in large burial mounds along with a treasure trove of artifacts that can be viewed in the museum. These include intricately craved sleds and carriages, plus many items used in daily life.
1: Oslo Opera House
Built in 2008, walking up the roof of the Oslo Opera House has become a must do activity while in Oslo. Clad in white Carrara marble, the Opera House evokes a glacier sliding into the fjord. A walk on the roof gives you a great view of the harbor, the bar code buildings and the emerging downtown Bjorvika area. It's free to walk on the roof and check out the lobby, but inside tours should be booked in advance.
Special Interest Sights
Some would have the Munch Museum in the top ten. My view is that National Gallery has enough top shelf Munch paintings to satisfy most visitors. But, if you're a Munch fan, you'll want to take the Metro to the Tøyen station to see more paintings.
The Ekeberg Sculpture Park is a new sight and a personal favorite. But because it's eclectic collection is dispersed through a large forested area, a visit here takes more time than most visitors can allot.
Downtown, the Nobel Peace Center has information about all the winners of the Nobel Peace Prize plus rotating exhibits about promoting peace. Lots of hands on exhibits makes this a good place to take kids.
After Shakespeare, Henrik Ibsen is the most produced playwright in the world. His last residence houses the Ibsen Museum.
The Royal Palace reigns above Oslo's main street, Karl Johan. You can walk around the outside for free at any time. Inside visits for the public are only during the summer and need to be booked in advance.
The main draw of the Astrup Fearnley Museum of Contemporary Art is the building itself. Designed by Renzo Piano, this landmark building with a sloping roof marks the entrance to Oslo's central harbor. A small outdoor sculpture park with a great view of the Oslo Fjord can be viewed for free at anytime. Inside, the museum has a permanent collection of big name contemporary artists, like Jeff Koons and Damien Hirst as well as rotating exhibits.