I flew from Toronto to Rome with a connection in Paris and I didn't have any issues with flying with my flute and clarinet.
Other than in Toronto, no one even did anything beyond asking me what it was. In Toronto they opened both my flute and clarinet case and kindly listened when I politely asked them not to open the cases upside down.
To cut down on any issues, I took out everything in my flute case except my thumb port and everything in my clarinet case except my mouthpiece and reed case. All my stands/tools/unopened reeds/swabs etc... I left in my checked luggage.
Musical Director on the Navigator of the Seas / Lead Alto Sax + Doubles in the Orchestra. :o)