Keeping your headshot relevant in our social media dominated world is important. Not only do you build in instant brand recognition, there is also a sense of trust your customers will feel when your photo accurately represents what you look like.
Professional businesses such as medical offices, attorney's offices, government agencies, etc., typically have a formal business attire requirement for their headshot sessions and have a set background. This creates continuity in the brand of the business.
For the creative, a comfortable, business casual attire will be acceptable for your headshot. These images may be taken in a more casual atmosphere on location such as out doors and in studio spaces.
If your style crosses back and forth between formal and the creative then you will want to have a mixed collection of headshot images to use for various media outlets.
These would work perfectly for Instagram, Facebook, Twitter, Tumblr, etc.