TSA policy allows musical instruments to pass through security as a carry-on. They are subject to the same screening as any other carry-on, including possibly being hand-checked. The owner is encouraged to be present if the instrument is hand-inspected. The TSA defers to carrier size limitations and specifically states that airlines may still refuse to allow certain instruments on board even if they have made it through security.
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Stated policies for various airlines regarding musical instruments as carry-ons are as follows:
Southwest - Guitars/basses need their own ticket or the instrument must be checked. Maximum carryon size is 24"x16"x10", no exceptions. [Invalid or Expired Link Removed].
American - Guitars/basses are allowed as an exception to carryon size restrictions provided the instrument can be made to fit in the overhead space or garment closet (this is shaky; if space runs out your bass will probably be the first to be checked). It counts as your one carryon bag.
American's policy in full.
Delta - Musical instruments must meet carry-on restrictions. Total linear size must be less than 45" (most guitars/basses will not be permitted as carry-ons under this restriction). You can buy your instrument its own seat.
Continental - Musical instruments can be carried on board provided the total linear size (L+W+H) is no more than 51 inches (Your average bass in a gig bag will likely be too large: the average bag is 40"+19"+3" = ~62") and the instrument weighs less than 40 lbs. A seat may be purchased for oversize instruments, but URB's are not allowed as cabin luggage.
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Northwest - Guitars may be carried on provided the max linear size does not exceed 45" (Most guitars/basses will not meet this restriction) and the item will fit in acceptable carryon stowage. Musical instruments count as carry-ons. You can also buy your instrument a ticket IF you do so in advance.
Northwest's policy in full.
US Airways/America West - All musical instruments with total linear size less than 75" may carry on. (This is the most lenient yet; most gig bags and hard cases are allowed under these rules).
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And for those travelling outside the U.S.:
Qantas - Carryons are limited to 45", or 73" for a NON-RIGID garment bag (your bass will probably not be considered either) and counts as a carry-on. Musical instruments exceeding this may be carried on given approval by Qantas on a case-by-case basis. You may need to buy the instrument its own seat. [Invalid or Expired Link Removed]
British Airways - Carryon limitations prohibit practically all guitars/basses (22" max longest side). No specific mention of musical instruments. Their "baggage tips" say not to pack valuables in checked luggage. No mention of cabin seat baggage for fragile items.
BA's policy in full
Air France - Standard hand baggage requirements prohibit most musical instruments (22" max on its longest side) by themselves, however musical instruments are apparently given an exception, with banjos specifically mentioned as an instrument that can be carried on with no prior arrangements. Guitars/basses may therefore be allowed.
Lufthansa - Guitars/basses may not be carried on (max longest dimension is 22" with the only exception being foldable garment bags). most flight cases however are allowed as checked baggage with no additional charge for bulky items.
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All Nippon Airways - Guitars/basses cannot be carried on (max length of longest dimension is 22"). Airline is not responsible for damage to musical instruments in checked baggage.
ANA's policy in full
Indian Airlines - Portable musical instruments not exceeding 45" in length are accepted as checked luggage. Carryon luggage is restricted to essential personal items only.
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Conclusion? If you plan to fly, get the flight case. It'll save you a lot of headaches as most airlines do not allow guitars or basses as carryons, and those that state in their guidelines that they will accept them still have the right to refuse any bag for any reason. The term "musical instruments" apparently does not yet refer to guitars/basses; the instruments that would be allowed under said rules are violins/violas, flutes/piccolos, clarinets, oboes, trumpets, and similarly-sized instruments.