Filipino is the Hispanized (or Anglicized) way of referring to both the people and the language in the Philippines. Note that it is also correct to say Filipino for a male and Filipina for a female. Never use or say Philippino, because that doesn't sound right.

Which is correct Filipino or Pilipino?

The language spoken in the Philippines is Pilipino (note the absence of the h). The people are Filipinos (men) and Filipinas, also without the h. Because it's a Spanish word. I don't think there are any Ph combinations in the Spanish language.

Is Filipino singular or plural?

The plural form of Filipino is Filipinos.