Apart from flying flies, wasps and mosquitoes (which I never have seen on stamps so far), the flying 'objects' I see around my home on a regular base are birds and airplanes (and leaves and maple seeds in autumn).
Dutch PostNL has started to issue a serie of four beautiful bird photo stamp sheets this year.
The third and fourth stamp sheet I hope to get in future. At this moment I can show you some flying birds from the first two sheets:
From the 'lentevogels' ('spring birds') the stork ('ooievaar' in Dutch):

and the black redstart (in Dutch: 'zwarte roodstaart' which literally means 'black redtail'),

And from the previous 'winter birds' sheet this greylag goose:

This flying bird I happened to receive yesterday, but it is in transit. The stamp is on someone else's chaincard, so I have to forward it. But fortunately I could make a scan in time :-)

Luckily some months ago I received one of my own chaincards back, themed 'aviation'. Here it is, after traveling from the Netherlands via Singapore, Thailand, Russia, back home, now showing several airplanes, and a helicopter:

See more interesting stamps, and wonderful variations on the theme 'flying' today's Sunday stamps and follow the links.