Bathroom Drain Venting
How about this:


Or, if permitted by code an air admittance valve (AAV). The AAV is a simple in wall or under sink vent that allows air to vent when needed but blocks water and sewer gas from escaping. Just remember, the AAV must be accessible and are usually covered by an access panel for inspection.

Option #2 is also fine. It's called a "Chicago Loop" and used for sinks on kitchen islands.

Option #3 wont allow the sink to properly vent causing drainage problems.

If that sink is near an exterior wall, you may vent it directly outside, code permitting of course.