All three terms refer to the same thing - table tennis table.
乒乓球桌 is a commonly used (1.58M results on Google, 2.60M results on Baidu) layman's term.
乒乓球台 is the technical term (both Baidu and Wikipedia use
this term) and is commonly used (2.14M results on Google, 4.19M results on Baidu) as well.
乒乓球案子 is rarely used (0.21M results on Google, 0.13M results on Baidu) from my own personal experience.
When used by itself,
桌 refers to a table.
台 refers to a platform or a stage.
案子 refers to a long table or a plank used as a make-shift table.
If you are discussing about table tennis in general, you can also use
球台 to refer to the table.