I never try to arrive at an event empty-handed. If it's a birthday party at a restaurant: a gift and/or card, plus the knowledge that I'll be paying for my own (and others if the bill is divided equally be the # of people attending) food.
If it's a bbq or house event, I'll usually ask the host if they'd like me to bring something. If I'm not that close to the hosts I won't ask, and I'll just bring napkins, sodas or the like. If I know the hosts well, I'll bring some food dish or whatever I'm asked to bring. If it's a dinner party at a home, I'll usually supply a bottle of wine or vodka or some part of the dessert.
LOL @ folks who expect gifts at their events. I decided long ago that I would never host an event (with the exection of a baby or bridal shower) where I plan the event with the expectation of supplementing the event with gifts/$ or donations. Any gifts, food or donations brought are thoughful and appreciated additions. I throw parties for myself, and invite folks simply to be a part my celebration. If I cannot afford to do it all with my own bank account, then I should scale back or not do it at all. That's probably why I've only held an event once at a place (a club) that folks had to pay at the door to get in. I felt bad about that and vowed to try and avoid that in the future.
In fact, I wonder if it's even good etiquette to "invite" someone to party with you at a dinner "they" will have to pay for (unless it's work related)? I wonder if it's more appropriate to say "please dine with us at "John Doe's Restaurant" on June 20 in honor of my birthday, rather than "inviting" someone to attend your "birthday party" at John Doe's to which you will not contribute any money or even much effort?