There are two flagpoles at the Kerbal Space Center. One is located in front of the Astronaut Complex while another is located at the Launch Pad. They show the default flag for the current save game. Flags are destructible if their base is hit hard enough, and cost 15,000 Funds each to repair.
The flagpole at the Astronaut Complex is technically a separate building and can be used to select the default flag for missions. The flagpole at the Launch Pad is not a separate structure; it is part of the Launch Pad and counts towards its repair cost if destroyed. To access them you must travel to them after launching a craft on the Launch Pad or Runway. Both of them are flying in version 0.90.0.
Somehow, crashing a plane into the flagpole will bring it to a complete halt.