There are two parts to the Palais Liechtenstein, one is the Garden Palace which is located in Alsergrund and the City Palace which is located in Bankgasse. There used to be a museum in both locations which had to close due to the Anschluss in 1938, though the museum reopened in 2004 but that too had to close due to low visitor numbers in 2011. You can still visit the palace however and there are special events and guided tours. You may find their website useful:
The Princely Family still own the palace but I'm not 100% sure if they regularly use it.
"She is a little angel and like her name, she brings sunshine even on cloudy days. From the bottom of our hearts, we would like to thank each and every one of you for your lovely best wishes for our daughter. She feels very loved". HRH Princess Madeleine, Duchess of Hälsingland and Gästrikland on her daughter, HRH Princess Leonore, Duchess of Gotland.