Hmm...ok I understand if the cat is in the scene, but what if there is no one else in the scene and it's just my character? Does the same thing apply? For example I have my character muttering to herself.
Yes. Ordinarily you want a new line of each new speaker, for clarity, but as you've pointed out, there's only one speaker.
I think you'd still use the "new line" rule, if only to avoid a massive, scary-looking paragraph that people will skip over.
For example:
Alma set herself down on the lumpy sofa. "They were here a minute ago, I swear," she muttered.
She cast her mind back to the last place she'd used them. "I had my copy of Reader's Digest... went through the
Pays to Increase your Word Power..." She'd gotten nine out of ten correct, Alma recalled with a satisfied smile.
Then she'd set her glasses down -
where? somewhere - and had gone outside to check the mail. Nothing today. "Typical." She plucked a loose thread on her sweater. Not that she'd expected otherwise: no one ever sent her a darned thing, even today...
"But it would've been nice," she said quietly. "It would've been nice."
I think the rule is probably break up lines into topics, or for impact.