Let's begin with the poor bloodhound. Most of the time when we see him, he's talking about what may be, or is, wrong with him, among other gloomy things. However, towards the end of the Pet Shop episode, he smiles as he talks about himself getting adopted by a nice family. At least he counts his blessings!
In the US series, the Buffalo talks about how he's gone through hard times, but his grandchildren make life worth living. The animation has a few tears roll down his face and he's clearly choked up in happiness.
Another example from the Pet Shop episode. Trixie and Captain Cuddlepuss are most often seen arguing with each other over something petty; however, when asked how much they're worth, Cuddlepuss says he wouldn't sell her; and Trixie wouldn't sell him, either. Aww.
Towards the end of the episode on Families ("The Brood"), the Pig gives a speech about how much he loved his father, how he saw him as a role model, and how he doesn't think his own parenting will ever measure up to his dad's. The animation has him rub a tear from his eye towards the end; you can definitely sense his mournful admiration. It's almost a Tear Jerker on top of that.