Very definite no from me, though I suspect some (very few) animals would be far better behaved than some of the humans I've encountered on planes over the years.

I sat next to a lady with a guide dog many years ago.  The dog's behaviour during the entire flight from boarding to disembarking was impeccable.

Having unnecessary animals on board would also be problematic in the event of an emergency evacuation.

Definitely no pets on planes...only exception, guide dogs.

Guide dogs and certified, trained companion animals only.

I don't have any problems about having pets on board flights, but do have concerns about the welfare of some dogs being flown at all. Dog breeds with the typical flat faces of pugs, boxers etc have so many problems with normal breathing that their lives can be at risk if transported by air. I understand there is at least one airline in the US that won't transport any of these dogs due to the health concerns - I'm sure a major concern for them is the possibility of being sued should the dog come to harm as well. I have flown different dogs and cats at various times without too many issues but prefer not to as I think the risk of them coming to harm when shut up in the hold is too high. There is also the risk of the airline losing the animals! This happened to my daughter-in-law's cat. I put the cat on a direct flight from Darwin to Sydney where her 'mum' was waiting to collect her when the plane arrived. The airline staff couldn't find her! It took all day for them to find her left in the luggage area of a shed. The poor little animal had been without food or water all day and was very distressed when she was eventually found and handed to her poor 'mum' who was beside herself with worry by this point. At the very least, I expect an animal to be safe if transported by air. These days we only travel by road if we need to take our dogs with us. Apart from all that, I do understand that there are some, maybe many people with allergies to some animals and it would be miserable for them to be beside someone's pet. It should be possible to have a specific section of the aircraft where animals could be contained though. This is done pretty successfully overseas and has been for many years. Even as a child in the UK in the 40's and 50's it was commonplace for dogs to travel on public transport - buses, trains etc- with their owners.

I don't like the idea of people bringing their animals into restaurants as well. It's ok to sit outside with them and a bowl of water, but definitely not inside. 

I cannot see a problem with pets travelling in a plane, after all we let children sit in food trolleys in supermarket and goodness knows where they have been with their feet before and then other people are expected to put their food inside the same trolley.