Can I take a selfie with my mobile phone for my PR or Citizenship application in Singapore?

Its upto you. Maybe asking somebody to take the photo with a plain colour wall behind is a better option.

The best option always will be to take a photograph from a professional photographer. Usually, you will find one in the shopping malls or the nearby market. These photographers are well aware of ICA requirements. Also, they have professional lighting available. They can give you photos on the spot and you can ask for a digital file in jpeg or png format.

Ensure to avoid following mistakes (as advised by ICA) while taking a photo, especially if taking it yourself.