I think that you mean that you are looking for a single fare for a Caribbean cruise don't you? Fred Olsen are advertising fares of around £115 per day at the moment, including air fare. There will, no doubt, be more about as the winter gets closer. If you are simply looking for the cheapest fares, then I would keep your eyes on Fred Olsen and MSC cruises. Expect what you pay for though! You could always choose the QM2? Happy cruising.
If you are looking for a cruise that will only have singles onboard then you are going to have very little to choose from. There may be some lines that have a cruise or 2 in their caribbean schedule that are designated as singles but I very much doubt they will be exclusively so, more likely to be geared more to singles and have more activities planned. if you are looking for a cruise suitable for a single person, then most are. I often cruise on my own and I'm rarely the only one. The big issue is that there are few ships with single cabins and the supllements for occupying a double cabin on your own vary a lot. the cruiseline i use is 80% for inside and outside cabins and 100% for balconies and suites. Some may be better than this, others more expensive. I'd call the number on this website and ask their advice on what are the better deals. You'll also be able to discuss with them what sort of cruise you are looking for (how many star, huge resort ship/smaller ship; formal/informal/ very lively party atmospher or not et etc)
Good luck on finding something. I find that cruising is far and away the best type of holiday to do on your own.