Segregated services/accommodations/etc weren't found generally throughout the United States. These laws were typically found in Southern states, which also tended to have the lowest foreign immigration during segregation. I can't find it at the moment, but I remember an earlier post about this issue, and it just seemed to be a case-by-case basis. Because of the racial classifications used at the time, locals would decide if a Chinese person was "Mongolian", for example, and segregate them with the non-white population accordingly.

It's worth noting that Latinos weren't always considered "non-white." As people of European descent speaking a European language, Latinos (generally Mexican-Americans) were considered "white" for a long time in much of the United States. The first interracial marriage struck down in the United States, for example, was the 1948 case Perez v. Sharp. A Mexican woman tried to marry a black man in California, but the anti-interracial marriage law prevented her from doing so, since she was white. The California law prevented whites from marrying "a Negro, mulatto, Mongolian or member of the Malay race", but the court held that the law was contrary to the Fourteenth Amendment. As you can imagine, these racial divisions were hardly helpful at a time when Chinese and Japanese citizens were a non-negligible percentage of Californians.

One last point, more directly on your question about Asian Americans. In the case legalizing "separate but equal", only one justice dissented from the case. While the dissent is famous for decrying incipient segregation statutes that discriminated against black citizens, it was less than stirring when it came to Chinese residents of the United States. In fact, Justice Harlan says that one of the reasons that segregation is so evil is that Chinese could share accommodations with whites while blacks were given second class treatment:

There is a race so different from our own that we do not permit those belonging to it to become citizens of the United States. Persons belonging to it are, with few exceptions, absolutely excluded from our country. I allude to the Chinese race. But, by the statute in question, a Chinaman can ride in the same passenger coach with white citizens of the United States, while citizens of the black race in Louisiana, many of whom, perhaps, risked their lives for the preservation of the Union, who are entitled, by law, to participate in the political control of the State and nation, who are not excluded, by law or by reason of their race, from public stations of any kind, and who have all the legal rights that belong to white citizens, are yet declared to be criminals, liable to imprisonment, if they ride in a public coach occupied by citizens of the white race

The fountains, restrooms, etc. were not labeled White Only and Black Only but rather White Only and Colored Only. Did the word colored mean anybody but white?

Follow-up... Are there any documented instances of* whites using 'colored' facilities as a means of protest? If so, were they beaten, shunned, etc?

Really interesting topic. Does anybody have any references for Asians other than Wikipedia?

Answers have been given for Latinos and Asian people. What about non-black people with colored skin, I'm thinking Indian/Desi people?

Since there are a lot of informed people here, can I ask if there's a good single book that covers how racial categorizations have evolved over time in America?