We had a Jack Russell that used to fly in the Jodel I once had a share in. She was fine.... looked a bit mystified (in a Gromit kind of way) when we took off, then settled down to sleep until we landed. We now have two Yorkies (dogs, not chocolate bars) - one of whom does just the same as the JackRussell. The other is a nervous wreck, even in the car. We tried 'em out in my 172 but I refused to take the nervous one up - just taxying around made him behave in a very panic-stricken way. The last thing I need is a dog throwing a wobbler when I'm flying.
And years ago, when I was an ATC cadet, one of the civvie instructors used to take his spaniel up on glider flights, sitting in the open-cockpit back of a T31 glider on his kap. You could hear the pooch barking all the way round the circuit. But it enjoyed itself. When they landed, it wanted to go back for more!