In this town, it's the shopping badge (spend 5mil one). Most of it was spent on getting Gracie Grace's furniture though.
In my old town, it was the Gardening badge. My newbie self didn't know the existence of permanent dirt patches, so I kept on watering a ton of flowers on a daily basis.
Wishy got a gold medal for going to a lot of K.K. Slider's acoustic guitar concerts.
Jirachi got a gold medal for playing the game for 500 hours.
These are actually the only gold medals I've gotten.