The thing to remember with assertiveness, is that you can't label actions, or other people, as they can pick that apart (and it is hostile, which is no better than being passive). So don't tell your MIL that she's rude, or that her actions are rude: she will just deny it, and then the conversation will become about that.
Instead, focus on "I" statements: what you feel about the rude action, and what you need instead. Because only you get to decide how you feel, and what you need, and they can't pick that apart. They can still fight against it (and will), but your position is your position. Assertiveness is about stating that position, and then letting the other party react as they will.
So:
"I find that comment hurtful. Don't compare me to the cat."
"I'm talking. Let me finish my sentence."
Don't say please or sorry.
Don't get into debates about whether an action is rude or not: State how you feel about it, and then give a direct order for alternative behaviour.
Since you asked, a real-life example that comes to mind is the following: my parents have a pretty dysfunctional relationship, where my mother is controlling and demeaning, and my father just takes it and whimpers. It's awful. She's stopped doing it to me since I went NC for a year and a half. I still don't enjoy witnessing it when she turns her aggression on others. So last Christmas when she was laying into my Dad, I said: "I don't want to hear this." They stopped. If they had carried on I was prepared to walk away.
I didn't get into an argument about whether my mother's actions were justified or not: clearly she thought they were. I just stated my own, unassailable position, which is that I personally don't want to witness it.
Another example: I left an abusive ex-h 5 years ago. We still had to see each other from time to time with lawyers present to deal with our house sale. This meant that he would try to engage me in conversation in the waiting room before my lawyer had arrived. So, whenever he said anything to me, I would respond: "I'm not interested." Again, that's unassailable: I am the only judge of whether I am interested in holding a conversation or not. I gave no reason and no apology, because my position, which involves only me, is not open for debate (you can't state positions for other people). It leaves no leeway for manipulative types to try to control the conversation. And it worked.