I wish I knew a story about Rav Scheinberg's righteousness, but the only story that comes to mind is a spooky one.
A girl - I'll call her Bridget - had met a young man and wished to marry him. "Don't," said all her teachers. "Don't," said all his teachers -- which tells you something. But Bridget went ahead and married the fellow, and soon realized his true nature, and they divorced.
Every time Bridget looked in the mirror, she hated seeing her head covered, because it reminded her of her unfortunate decision. She asked Rebbetzin H. to take her to ask a rav whether it was really necessary for her to cover her hair now that she was divorced.
Rebbetzin H. (who told us this story) took her to a rav who spoke English: Rav Scheinberg.
"Why did you bring her to me?" asked Rav Scheinberg. "You know how I pasken."
He told Bridget that she did have to cover her hair. Bridget was displeased.
He suggested a way that she could do it prettily. Bridget would have none of it.
Ultimately, he gave her a blessing - what it was I can't remember; I think it was something like that her covering her hair should lead her to good things.
While in the Scheinbergs' apartment, Bridget struck up a conversation with Rebbetzin Shain, his sister-in-law, and in the end she wound up a regular companion of the Rebbetzin.
A mother and a son - I'll call him Yitzchak - were sitting on a bus, having a conversation.
"You've met so many nice girls," said Yitzchak's mother. "Why can't you settle down and get married?"
"I want a very special girl," said Yitzchak. "See that married woman helping Rebbetzin Shain - how solicitous she is of the elderly rebbetzin's needs. I would like to marry a girl like that." This story takes place in a society in which Yitzchak would never look at an unmarried girl on a bus - he referenced the stranger only because her covered hair indicated that she was married.
"Rebbetzin?" asked Yitzchak's mother, some hours later. "That woman who accompanied you on the bus today -- does she have, perchance, a younger sister?"
"Why do you ask?" asked Rebbetzin Shain.
...and a few months later, there was a wedding.
And they all lived happily ever after.