Very often in public place, I use sticker from the most famous and public radio in France. Everybody think that i'm a journalist, i never had problems, expect people who tried to speak in my microphone...

In public place like airport, the best is to use 2 DPA 4060 or 4061 or... and stick it on your bag near your soulders.

Every time I go through airport security with my mbox, I get stopped and they swab it and put it in that explosives testing machine. For about a year and a half, every time they did that, it came up positive for dynamite. The first time this happened, I was put in this holding area and the TSA guys called an FBI agent down who questioned me for like 30 minutes before they let me know. Still have no idea why this happened. The only thing I could think of is that I lived a block away from a building demo in NYC at the time.

I've also been asked to demonstrate the functionality of my Sanken CSS-5 a few times to prove it wasn't a weapon, since it looks quite like a gun with it's pistol grip on.

I've been pulled over before while handing a fishing rod with a mic on it hanging 10 feet off the back of a panel van to get a medium perspective exhaust sound.

I've been chased away from train tracks while trying to use a contact mic to get rail sounds

That's all I can think of right now. I've heard some great stories about authorities before. Hopefully we'll get some good ones here!

It wasn't me, but when I worked on the Quantum of Solace game, one of our sound designers went to the LA airport (LAX) to get ambiences. He was just walking the streets near the airport, not even on airport grounds, but it didn't take long for airport security to stop him, check his ID, and tell him to move on. That's the reason he was the one recording there and not me :)

In the US, only with photography so far. Photogs often carry small cards with the legal guidelines around taking photos (er, sorry, making photos) in public places in case they get hassled. This doesn't win the argument every time, but it helps you from backing down every time. The trick is that often what appears to be a public space isn't, and people's paranoia makes them think that standing on public land but shooting a private building is illegal...it's not. Depending on the context, though, doing so can seem creepy, invasive, or innocent, and you'll probably be asked to stop anyway, legal or not. (Photographing PEOPLE has a whole different set of laws and issues.)

But back to the audio side of things. The other industry that knows a lot about audio laws are radio journalists. A great URL to reference is the "Can We Tape" guide to taping conversations, in person and on the phone. Turns out a lot of it is like photographing people. This suggests that general audio recording will probably follow the legal guidelines of both photography and videography, as a general rule, especially since surreptitious audio picked up by a video camera is often covered under wiretapping and eavesdropping laws. There will be exceptions, and I'm no lawyer, so take my opinion at your own risk. :-) Even in the US, every jurisdiction is different in what laws it has and how they're interpreted.

But we, as sound recordists, mixers, and sound designers, are often recording all sorts of random crap. Our actions are very hard to interpret when compared to photography, and it's our responsibility to be sensitive to this without being bullied by those with ignorance of the law. Showing credentials is important, getting permission on private property is vital. The rest sadly appears legally murky.

in Italy, I'm quite often mistaken for someone from the NHS controlling noise levels...it happened when I was recording sounds from a porhpyry construction site. In 15 minutes I've been quite angrly addressed by: the major, the council first engineer, the council major, everybody screaming that the construction site noise level was not THEIR responsibility. Once explained, the offered my a drink and apologised.

At school, while we were studying microphones, one day we had to go out and record some things. So we go to this outdoor promenade thing with fountains and right next to it is a park. We set up two microphones and stands and start recording into an MBOX.
That went fine.. We decided to move and try some new sounds and we had barely got the first stand set up when what looked like a security guard out of uniform and coming back from lunch walks up to us and asks for a permit and says we'd have to go and get one from the main office if we were to stay and record. Thats all he said so we packed up and walked down the block to record somewhere else.

I had a wee spot of trouble with local authorities on a video shoot once...the full story is on my blog; rather than re-type it or copy/paste I'll just link to it. Enjoy!

i was once stopped from recording ambience at the train station in York (UK). I was using a hand-held recorder (H4n i think) and being reasonably subtle about it, but station security spotted me and i was told to stop and that I needed a permit. which i'm pretty sure is not true.