How or where do we see sexism in children's literatureProvide examples
It would be difficult to cover all of children's literature. In fairytales, young females were always abused and endangered by older women ("stepmothers") and needed a handsome prince to save them. Snow White is housekeeping for seven "men" who go off to work, and they love her for her care of them, and protect her. Not surprising that this image is that of the "liberation" of women in the 1980s.
Modern day literature for children doesn't seem (as has been my experience) to create these rigid stereotypes. There are probably two reason. First, society is not looking for daughters to be taught to stay home and keep house, or to be overly dependent upon a man: today households often require two incomes. Secondly, authors want to appeal to readers of both genders. In Skellig, by David Almond, for instance, both Michael and Mina are "different," but each in his or her own way is not only smart, but kind and generous. It's a win-win situation. Both are heroes: the novel engages boys and girls.
The "wicked stepmother" is a frequent archetypal character, not only in Cinderalla but in many other fairy tales. There don't seem to be nearly as many "wicked stepfathers," although the reasons for this do not seem especially obvious. Perhaps it is because women, in the past, were traditionally the persons who dealt most frequently with children. Therefore, a wicked mother figure might have been a character who seemed more relevant to a child. In any case, here are some leads for you about wicked stepmothers:
I see a good deal of gender role stereotypes in children's literature. Most of the female characters are portrayed as helpless damsels in distress waiting to be rescued by the strong, independent male character. Sometimes it is more simple than that. Even Dr. Sues falls into gender role stereotypes. In his book One Fish Two Fish there is a page that says "girls who like to brush and comb should have a pet like this at home" and another page about a little boy who likes to box. It places male and female children into distinct character roles which they may or may not fit into. A lot of children's literature falls into this sort of gender role stereotyping.
I think we see less of this these days. After all, One Fish, Two Fish with the boy boxing his Gox in yellow Gox box sox is from 51 years ago so it's not exactly modern. We see much more in the way of assertive and independent girls now in series like the American Girl series and even comparative trash like all the Daisy Meadows fairy books. Nancy Drew has been updated and can take care of herself physically. We still do have the old "fairy tales" that portray sexism, but that is becoming much more rare in contemporary books.
I think there is some value in recognizing stereotypes such as the sexist roles that are rampant in fairy tales. It is important to make sure children understand that these are not the only appropriate ways for boys or girls to act.
However, #5 makes a good point, too. There is no need to beat the sexism concern into the ground and destroy the enjoyment of the literature in the process. Yes, Little Red Riding Hood is a girl and she turns out to be the heroine, but let's enjoy the story for its own sake!
You could observe the mothering instinct and subservience of female characters in stories such as Snow White. She is literally running for her life when the huntsman tells her to run and then comes upon the dwarves' messy cabin where she immediately gets to work cleaning it up, even before she takes a nap! She is valuable to the dwarves because she can cook, clean and sing for them, so they keep her around and come to love and take care of her.
As other editors have commented, most examples of sexism in literature for children involves the way in which roles are ascribed to certain genders and are presented as being "natural," such as girls being taught how to cook and clean, as in the case of Cinderella. There are unfortunately plenty of examples of how girls are given caring, nurturing roles and boys are more reckless and adventurous.
This is such a 21st century question! Why must we criticize literature by modern liberal/politically correct standards so often? There is a deconstructism that endangers the joys of reading fables and fairy tales. For, by searching for feminism, sexism, etc. we lose what the original fables, fairy tales, and legends were really about.
To me, the classic story "Cinderella" is a perfect example. Cinderella is victimized over and over again. She waits for her prince to rescue her. She does not take action to rescue herself, and she is helpless without him. This sends the message to girls to wait for the prince to save them.