Catelyn claims to accept that men of Ned’s rank often father bastards and provide for them.
But, she considers it an insult that Ned (a) won’t tell her who Jon’s mother is (though she believes it’s Ashara Dayne) and (b) treats him better than she believes is reasonable, specifically by keeping him at Winterfell (it’s a particular affront that the infant Jon got there before she did) and calling him “son” in front of others. She believes that bastards should be “out of sight” and that Ned’s refusal to send Jon away is evidence that he loves Jon’s mother more than her (while seeming to give herself a pass on having originally loved Ned’s brother Brandon). She’s also upset that Jon looks more like Ned than her own sons do (Arya doesn’t count I guess).
Is this reasonable?
If Catelyn were in Ashara’s presumed place, would she want the man who impregnated her broadcasting her indiscretions, even to immediately family? I personally don’t care for relationships where total surrender of privacy is demanded a proof of love. On the other hand, this secret is a significant enough one that Catelyn is not completely unreasonable in her desire to know.
On the matter of being more of a father to Jon than Cat thinks is reasonable: Ned is not unique in keeping a bastard at home. “Bastard Walder” lives at the Twins. Aurane Waters is called “the Bastard of Driftmark,” suggesting he lives there, and seems to have had a high-class upbringing. Joy Hill appears to live with the Lannisters, and is valued enough to be used as a token of marriage alliance with the Westerlings, who were at that moment a rather politically important subordinate house. Since we don’t have much direct knowledge of these characters, however, I’ll give Catelyn the benefit of the doubt for her statement that Ned’s closeness to his bastard is unusual in their society.
Robert’s bastards are treated more in the manner Catelyn would approve of. Most appear to be largely unknown to their father. The exception is Edric Storm, who is acknowledged because his mother is noble (as Ashara Dayne is). But he still lives far away from his father.
Overall, by the standards of her society Catelyn does have some reasonable basis for complaint. But her grudge seems more bitter than is warranted, and in any case should be directed at Ned and not at Jon. (And her idea of kicking a fourteen-year-old whose overt behavior, so far as we can see, has been nothing but self-effacing and helpful to the family, out of the only home he’s ever known, is just cruel.) If she had evidence, or even believed, that Ned preferred Jon to Robb, her attitude might be more understandable (if not more moral), but in all her POV’s I don’t remember her thinking this (we’ll see if my memory is wrong).
(On the other hand, Ned is a bit dense (as he so often is!) in not realizing that however cruel people at court would be to Jon, Catelyn without Ned’s influence would be even crueler. He’s also completely unaware of Jon’s hidden discontent, being “shocked” to learn that Jon would consider joining the Night’s Watch.)
And of course, here’s our second big piece of R+L=J evidence (thought not big enough for Catelyn to pick up on): Ned calls Jon “my blood” rather than “my son” or “my child.”