Best resources for finding an apartment in Lincoln Square, New York City?

I would try Craigslist first in the "no-fee" or "by owner" section so you can try to avoid a broker fee (which is 10% of a year's rent). If you can pay for a broker, you'll find an apartment pretty quickly. I would use Citihabitats or Bond. They're the most reputable -- I used Citihabitats when I first moved to New York and I got a great apartment. Sadly, brokers do rule the roost as far as New York rentals, but it is definitely possible to find one without paying a fee.