I've been traveling single for the last five years or so, and I do it a lot.
wrp96 summarized it well - complete freedom to do whatever whenever, but also times when you wished you had a companion.
The worst part is the 100% single supplement, although the VTG site has a good list of low single supplement cruises - there are more than you might think - seems like Carnival and Princess are the mid-price cruise lines that appear most.
Once you get up to a certain level on Royal (340 points?), they reduce the single supplement to 150% on all of their cruises - that has made my selection process a lot easier. However,
always ask the cruise line if they have a "single rate" - I have had good luck many times, usually on longer (especially transatlantic) cruises - supplements have run 20%-30%, although once there was none.
Most cruise lines run "single" events (lunch, happy hour, or late evening), although I suggest going to the 1st/2nd day events - people tend to diminish after that.
If you're single, so probably will be your tablemates, and many of those do not show up (as noted in a post above), instead opting for the buffet. If you end up with an empty table, talk to the dining room manager and ask to join another table - have met many people that way.
Still, there are a lot of "frequent cruisers" that repeat often - probably 7 or 8 people that I knew from previous cruises on my last transatlantic trip on the Allure.
BTW, once nice perk about Royal - if you do book as a single, you get double points, which gets you higher status quicker. I believe Celebrity may have adopted that policy as well recently, but that's why I switched to Royal some time back.