I am a foreigner and my student visa expired last month. I do have an American driver's license valid until the end of next year. Can I still fly inside the US without being afraid they will check my papers and try to get me out of the country? I haven't used any of my European IDs or passport in 2 years since I got the US ID, so I haven't been showing them off. I've been feeling pretty secure with my license. What do you think?
To answer your question literally, you can certainly fly within the US with an expired visa. Your driver's license or your passport are both acceptable IDs according to TSA rules and if you show your license I don't think anybody will ask to see any other document. It might even be possible to fly without any of the IDs on the list, see I'm a foreigner, can I fly inside USA with my USA visa if my passport expires? and Can I fly domestically in the US using my university ID?
Now, if you're also out of status and staying illegally in the US, I don't know if you run a higher risk of being found out by taking a domestic flight. But an expired visa in itself is no big deal and it is not illegal to stay within the US without a valid visa (unlike European visas, US visas are only for entry and not about your right to stay in the country).
You should note that at some airports and roads/trains near border areas, there are occasional random stops of passengers. If your english is flawless, you may be able to get away with just showing your driver's license, but if there is suspicion of your immigration status, you may be asked to prove your status.
More information can be found by searching for "DHS interior checkpoints" or "Border patrol interior checkpoints"