I'm trying to figure out a way to handle a situation where an abusive neighbor has been shouting all sorts of obscene filth at her neighbor. She was ticketed under 42.01(a)(1), but we had to dismiss when we looked at the "public place" element before we got to trial. I can't really find anything we can do to make it stick or shut this woman up. It's not 42.07 harassment under any of those definitions. So what can we do if the offender never leaves her own property or makes an actual threat? Anything?

If it never gets beyond foul language then I don't think there is any criminal law option.

Depending on the full context of what the person is saying it could possibly be enough to go request a peace bond under CCP Art. 7. That would be the lowest threshold required in terms of what has to be threatened and the burden of proof required.

I had an intake call once where a feud prompted a homeowner to teach his parrot to cuss out his neighbor all day and night.

Shouting? How about a municipal noise ordinance?

Austin prohibits unreasonable noise between the hours of 10:30 p.m. and 7 a.m. The rule is enforced against private property when rock-n-roll bands play at night... and still they call it "Live Music Capital of the World!"

would 42.01 (a)(5) fit?

Unfortunately, it's not in a city limits, or else the city could pick it up. And I don't think 42.01(a)(5) fits, because unreasonable noises are addressed in (c)(2). It's more about the decibel level than what's being said.