A Hostelling International (HI; www.hihostels.com) membership card will get you a 15% discount at youth hostels.
Honnør (senior) discounts are available to those aged 67 years or over for admission to museums, public pools, transport etc. The discounted price usually amounts to 75% of the full price. You don't require a special card, but those who look particularly youthful may, apart from enjoying the compliment, need proof of their age to qualify.
Student discounts are often available (usually 75% of the normal fee). You will need some kind of identification (eg an International Student Identity Card; www.isic.org) to prove student status. Some travellers have reported being refused access with their normal university cards (unless it's from a Norwegian university), so the ISIC card is a good investment. It can provide discounts on many forms of transport (including airlines, international ferries and local public transport) and in some internet cafes, reduced or free admission to museums and sights, and cheap meals in some student restaurants.