Like vkniebes I prefer to use the hard surface carriers. I have only had one pup clawing at the soft side carrier, but the potential is there to have problems. However.........I cannot speak about other planes, but in the case of my plane the soft side carriers are my only option for large dogs like Dobies. I cannot get a large hard sided carrier through the door and in place with out front seat removal, but I can insert a collapsed soft side carrier into the rear and then set it up and load the pup. Without that ability I would be removing seats which would be a real pain.
However, for those trying to decide which carriers work best, here are some observations. Soft side carriers as mentioned simplify loading the carriers, but if the animal relieves himself or vomits they are not as easy to clean. I use a pressure washer at home to clean the carrier if there is a need. Any attempt to secure a soft side carrier is an illusion. It can be strapped in place but under any loads at all the carrier will collapse and the securing straps will go slack. Recognize that risk. In my case I only use large carriers and they are placed on the cabin floor directly behind the two front seats.
Hard side carriers are easily cleaned. If there is an accident I hose them off. They also can be nested by breaking them down and putting the two halves nested together. If I cannot clean them, at least I can put them in the baggage area which is separate from the cabin. That feature also allows me to carry more carriers than I need so I can juggle various sizes to fit all pups. Hard side carriers are easily secured and I feel confident they will not go flying around. I run my straps over the top, through the handles and then cinch them down with cargo straps secured to the seat anchors.
I have no experience with the straps that go around the pup and then attach to the seat belt, but I watched a fellow PNP pilot load and secure a large dog in his rear seat using the strap and he said the pup rode well and it appeard the pup was secured.
Whatever you choose is likely based on the plane and the animals being carried, but one thing is for sure. No animals should be loose in the cabin. That is a recipe for trouble.