Some lines including Cunard; Fred Olsen and Thomson to name 3 and there are others but to get the best info call the cruise desk on 0800 091 1629 and they will be able to give you prices but occupying a double you could look at 20 to 70% of a price hike, so worth enquiring further. Most ships who offer singles do organise get togethers for all those who want to join in. NCL did have an arrangement to pair up passengers, not sure if this still is in place. They put you in touch months before so you know a lot about the person before you join the ship.
As Blyth says some line do have single cabins but a lot don't. Occupying a double can be as much as 100% or more supplement but some are more reasonable. The best cabins at the most popular times are obviously going to have higher supplements, I often cruise on my own abd find the best value for length of cruise are the repositioning voyages 9where ships leave or return to Europe to say the caribbean, South America, South Africa) I've sometimes paid more for a week in the med than I have for a 19 day transatlantic sailing with the same ship and cabin type. I'd call the number that Blyth suggests and get an idea of what are the best deals to be had for single occupancy, then decide on location and the type of cruise you want (eg something very lively and informal or something more formal or whatever) Good luck with your search.
P.S, booking very early or late can often mean big discounts which partially offsets the supplement.