What are specific examples that show that the children are not paying attention to Atticus’ advice about their actions toward other people?
In Chapter 9, Atticus tells Scout that he'll be defending Tom Robinson and explains that people might say bad things about him in her presence:
You might hear some ugly talk about it at school, but do one thing for me if you will: you just hold your head high and keep those fists down. No matter what anybody says to you, don't you let 'em get your goat. Try fighting with your head for a change...it's a good one, even if it does resist learning.
Later in the chapter, when the family visits Finch's landing, Scout "forgets" Atticus's advice. When her cousin, Francis, taunts her with regard to Atticus's defense of Tom Robinson, she attacks him and splits her knuckles on his teeth.
As clairewait notes, Atticus's attack on Walter Cunningham is another example of Scout's impulsivity. Because Atticus knows his daughter's short temper, and because he knows that the citizens of Maycomb say very ugly things in certain situations (like the one he's entering into with Tom Robinson), he continually advises Scout to control herself. As she is young and passionate about what she perceives is right and wrong, though, it's sometimes hard for her to control herself.
I don't think you can count Scout's incident with Miss Caroline on the first day of school - as this is really Atticus' first major sit-down lesson in considering things from other people's points of view.
However, the fight with Walter Cunningham which lead to supper with the Finches (and Scout laughing at his putting molasses all over his food) could be considered an example of Scout forgetting the lesson.
The best example, I think, is fhe "Boo Radley Game" - not only are the children making fun of Boo by acting out all the gossip stories about him and his family, but they know that what they are doing is wrong because they hide it from Atticus. They basically lie to him - and act surprised and ashamed when they find out he has known all along. It is a little more acceptable for Scout and Dill - because of their ages - but this is pretty unacceptable for Jem. He, of the three of them, should know better, and he seems to be the one heading the entire thing up.
Atticus specifically told the children to be kind to Mrs. Henry Dubose and explained that she is an old lady and doesn't understand much of what is going on. He specifically told the children not to retaliate when they hear dissenting discussion about him.
When the kids were going to town to buy toys on Saturday in chapter 11, Mrs. Dubose unleashed her worst on them. She told them what bad kids they were and called Atticus a name. At the time the kids just kept walking, but on the way back when she wasn't on the porch anymore, Jem let loose cutting the heads off all of her camellia plants.
This was blatant refusal to accept Atticus' advice about how they should act in front of other people.
I would say that for the majority of the book, they follow his directions.