Can I just point out the first time that he betrays Sansa Stark:
When he arranges the murder of Joffrey, he uses Sansa to deliver the poison to the dinner in her necklace. As a result, he framed both Sansa and Tyrion for the murder of Joffrey. This forced Sansa to flee King's Landing in his company. And we know that the reason why he wanted Sansa in his control was to marry her as a substitute for her mother. So this is what he does to people that he likes (at least as much as he likes anyone).
He murdered the jester, even though he didn't need to do so. He could have just paid the jester and taken him along as well. The only mild inconvenience there is that the jester might have later told someone else about Littlefinger's role in the Purple Wedding and Sansa's subsequent departure.
Later, he betrays her again. As she was reluctant to marry him, he sold her to the Boltons for Ramsay to marry. Can we seriously believe that he didn't know what Ramsay's pleasures were? Littlefinger knew that information about everyone. That also betrayed the Lannisters, but that's much easier to excuse. The Lannisters aren't exactly the type to draw loyalty to themselves. Littlefinger later betrayed Ramsay, but not soon enough to avoid Ramsay practicing his cruelty on Sansa.
Causing a split between Sansa and Arya is actually rather mild on his list of betrayals.
So why he is always called an untrustworthy person by everyone?
He promises Catelyn to help protect Ned Stark. Then he immediately gives Ned information that will get him killed, setting him on the path to Joffrey's true parentage. It's also worth noting that this ultimately gets Catelyn killed as well. Ned's proclamation starts the War of the Five Kings, and Catelyn dies during that war.
The ultimate reason why he is considered untrustworthy is that he will go back on any promise if it advantages him.
He has saved Jon Snow in the Battle of the Bastards, he just loves Catelyn Stark, and perhaps now Sansa.
If he loves Sansa, it might be better to be one of his enemies. He frames her for murder and then sells her to her future rapist.
He betrays his promise to Catelyn, who makes the mistake of not only trusting him but urging Ned to trust him as well.
And in order to get back to his plan of marrying Sansa, he needs Ramsay out of the way. He didn't save Jon Snow so much as he came to see Ramsay killed. He wants Sansa to take the throne in the North, so he can be king of the North.
I will give him this. If Sansa's choices were Ramsay and Littlefinger, Littlefinger is clearly the better choice. He is seldom cruel just to be cruel. He is cruel to his own advantage. Ramsay enjoyed being cruel. Littlefinger just doesn't care. But not being quite as evil as Ramsay is a pretty low bar that pretty much everyone except Joffrey meets.