Looking on the internet deeply has found these results:

public-schools is the most popular phrase on the web.

public schools 176,600,000 results on the web	public-schools 176,800,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
Nice neighborhood, and I hear the public schools are great. Who is there that has not jeered at the House of Lords, the military caste, the Royal Family, the public schools, the hunting and shooting set, the horrors of a country society? There's nothing wrong with public schools. I'm a big advocate for public schools. I can't stand the stench of public schools. Okay, our public schools have failed you. I work for the public schools, Stephen. But you get that in Cleveland public schools.	It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

public schools

176,600,000 results on the web

public-schools

176,800,000 results on the web

More popular!

Nice neighborhood, and I hear the public schools are great.
Who is there that has not jeered at the House of Lords, the military caste, the Royal Family, the public schools, the hunting and shooting set, the horrors of a country society?
There's nothing wrong with public schools.
I'm a big advocate for public schools.
I can't stand the stench of public schools.
Okay, our public schools have failed you.
I work for the public schools, Stephen.
But you get that in Cleveland public schools.

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.