What type of dog should not necessarily be trusted at all times?
Greyhounds. Not because they're aggressive or mean necessarily, but more that their instincts are more likely to get themselves and other housepets killed.
My mother received very specific instructions about keeping one as a pet as the one she adopted had been part of a racetrack that retired her due to failing the very high standards to compete. What she had been told was that a breed like this tends to live their lives like professional athletes. All their needs are taken care of but they're not given the same lives as a beloved pet to play and just socialize. They're given a regimented schedule where they are fed a specific amount, they are let out at specific times for bathroom breaks and kept in their kennels when not running. They've been sheltered to the point they would not know to rein themselves in from overeating or running into traffic. The instinct to run is bred into them. Every time they're outside, their hunting instincts kick in and they may forget that the cat or perhaps a small dog they recognize as family indoors is not prey. Ours can never be let off a leash or trusted to run free without a fence. We have to be careful not to spook her when indoors as well and she cannot be given too much food or she will become overweight quickly. Every meal requires that she be taken outside a half hour after because of her rapid metabolism and she often needs massive bedding because her long legs are unsuited for sleeping on beds and couches (doesn't stop her from trying to climb or lay on them).
I one time made a mistake of trusting her in our back yard, thinking she'd be safe in a fenced in area with the pond. It was nearing the end of winter and the snow hadn't quite melted off the ground. Sheldon, our Jack Russell/italian greyhound mix, decided to be an imp and run off instead of coming inside the house. He raced to the other side of the pond where the hill we lived on went up the bank steeply and Paris, the greyhound mom adopted, raced out to follow him. She tiptoed gingerly here and there and the moment I went to go pick up Sheldon, I heard a splash. Paris had jumped into the middle of the pond and while she was doing well swimming at first, she wasn't coming to shore like I thought she would. Two minutes in, her head starts to go under and I'm headed in like a moron to try to save her.
Now, winter hasn't broken just yet.
So the water is freezing cold and while it's a short distance, it was also deep. Paris was a small dog for a greyhound but still a large dog when trying to drag her to shore. I had only succeeded in yanking her by the collar to move to the bank and it was sheer dumb luck that she could and did because I was quickly running out of stamina in a ridiculously short amount of time. I make it just enough to stand up in the water, Sheldon is staring like a child who realizes that things have gotten way out of hand and Paris is running off with a very thin, very wet coat. I still couldn't do anything but stand and breathe from the exhaustion for several minutes. By the time I slog back to shore, barefoot in the snow, office clothes soaking wet and heavy, I'm pointing to the house and Sheldon meekly walks himself back without any fuss while staying with me. Paris herself is already waiting at the door and it takes us both a good hour to dry off and have all the towels washing with my clothes.
To this day, none of us have any idea what possessed that dog to jump into the pond. What we can be grateful for is that Paris never tried to do that again and in fact actively avoids it now. Still didn't stop mom from putting up a fence to keep her from jumping into the pond again. Still, she's a sweetheart, with a sassy personality and loves the home she's staying in. Mom doesn't regret adopting her and it's clear she's a good fit for Paris' needs.