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Certainly you can and many do. You will have to pay a premium for single occupancy. This will vary from zero (yes, there are a few around) to 100% depending on the cruise.
Yes of course! Its probably one of the safest ways to travel. I also believe there are specific cruises for single people but they do have many times special meetings for single people to meet up and get together on board most ships. The only downside is the extra expense as all the cabins are priced based on two people sharing.
You can request to dine with other single people if you like on P&O
On P&O there is a coffee morning for people travelling on their own on the second day of the cruise.
People are very friendly to each other on cruises and there are always events in which to socialise with other people but equally people will leave you alone if you want your space.
With P&O and the other ships that require you to sit down at dinner with the same people every night its a very easy way to strike up aquaintances so it means that you can go off and do your own thing in the day then meet up for dinne.