How to determine if a polygon shapefile is multipart or singlepart?

For ArcGIS v. 10.1 and older I'd create new field of integer type and populate it using:

 !Shape!.partCount

Note parser is set to Python.

It is much faster with Add Geometry Attributes tool available in newer versions of ArcGIS. Tool creates required field for you. This is how attributes on offer look like for multipart point input:

I found this tool very useful in general, e.g. see option selected below in the list available for polygon input: