For dogs, like many other species, early experiences are critical for normal social development, and it is pretty well known that puppies have the best chance for normal social development if they are allowed to be with their littermates for 7-8weeks at least.
It is really the exception for puppies not to have littermates or not to get to be with them for at least these few weeks. However, singleton puppies do happen, and they do tend to have issues. If you ever meet a dog named Solo, Uno, or Only, the first question to ask is whether the dog was the only puppy in its litter, because if so, there is a suite of problems that may exist.
Of course, you can be wrong about these names. I once wrote a magazine article for The BARK that mentioned a dog named Solo who had some serious behavior issues, and I thought at first he must have been a singleton. However, in researching the story, I learned that the dog came from a litter of several puppies and was named after the Solo River in Indonesia where fossils of Homo erectus were first found.
In a typical litter of three to twelve puppies, there is constant physical contact. The puppies crawl all over each other, and they are used to the warmth, the contact, the interruptions, and the movement that result from being in a pile of dogs.
The problems that singleton puppies are prone to having are the result of not being raised in this standard puppy environment. Typical problems in singletons are lack of bite inhibition, being unable to get out of trouble calmly and graciously, an inability to diffuse social tension, inability to handle frustration, lack of social skills, lack of impulse control, and touch sensitivity.
If you find out about a singleton puppy early -- anytime before the puppy heads to its new home particularly, there are things that can be done. Be sure to work on teaching bite inhibition early and often, and handle the puppy a lot to avoid issues with touch sensitivity. Any gentle, regular handling is likely to help. Push the puppy off the nipple once or twice a feeding to get the puppy used to interruptions and handling the resulting frustration. Have the puppy spend time with puppies of the same age a lot and as early as possible.
If at all possible, consider raising the puppy with another litter. Getting to spend a lot of time with another litter lets a singleton puppy have a more typical or normal experience as a young puppy. The play time that puppies spend with each other goes a long way towards teaching puppies many of their social skills, including bite inhibition, frustration tolerance, impulse control, self control, and the ability to be flexible in all sorts of social interactions.
The adorable play between puppies, which is so enjoyable to watch, is anything but light-hearted frivolous behavior -- it provides puppies the foundation for normal, healthy social behavior as adults in many contexts and is a critical part of a puppy's development and education.
I knew a singleton Irish Water Spaniel that I met at age two. He was full of himself, had no frustration tolerance, little self control and almost no impulse control. He did, by the way, show beautifully in the ring! His issues with frustration and control led to leash aggression with other dogs. His amazing owner, who had actually bred him, was able to turn him around, but it was a huge project. The next litter from the same female was also a solo puppy who turned out fine and totally normal, except for being a bit large for the breed, which is not unusual for singleton puppies. The owner did everything right with her second solo puppy. She raised this puppy with a Lab litter that was only a few days different in age than her puppy, and did everything else I advised.
She did end up spaying the breeding female, figuring that once could be a fluke, but that since it happened a second time, there was too high a risk of it happening again. This second singleton puppy, benefiting from all the owner did to help her, was in no way behaviorally like most singletons. She turned out completely normal from a behavioral perspective, despite an unusual beginning and this is an amazing accomplishment.