I think almost everybody is single after midnight on New Year's Eve. For that matter on most weekends too. (joke, kind of - but be careful!)
That being said, if you start the night out on your own, probably best to do it in town and to go to the usual mingle-spots rather than the romantic two-for-dinner ones. Fidos usually tops the list of casual fun bars with food and music, Pedro's probably has a party happening with single travelers too ..... and they serve excellent pizza. Holiday Hotel's bar and restaurant are probably going to be fun - and that crazy beach place (Loco Vida?) will definitely have something loud and wild going on. Follow the noise and music and you will find plenty happening.
I wish you happy new year now - I'll be fast asleep well before the calendar turns a page. Waking up without a hangover to view a sunrise on January One is my idea of welcoming in a new year.
That's something you can do with or without a partner.