Are mixed geometry types allowed in shapefiles?

Answer can be found from the official specification http://www.esri.com/library/whitepapers/pdfs/shapefile.pdf.

All the non-Null shapes in a shapefile are required to be of the same shape type.  The
values for shape type are as follows:
Value                                 Shape Type
0                                     Null Shape
1                                     Point
3                                     PolyLine
5                                     Polygon
8                                     MultiPoint
11                                    PointZ
13                                    PolyLineZ
15                                    PolygonZ
18                                    MultiPointZ
21                                    PointM
23                                    PolyLineM
25                                    PolygonM
28                                    MultiPointM
31                                    MultiPatch

Shape types not specified above (2, 4, 6, etc., and up to 33) are reserved for future use.
Currently, shapefiles are restricted to contain the same type of shape as specified above.
In the future, shapefiles may be allowed to contain more than one shape type.  If mixed
shape types are implemented, the shape type field in the header will flag the file as such.

Notice also that points and multipoints can't be saved into same shapefile while lines and polylines can, as well as polygons and multipolygons.

A shapefile does not support mixed geometry. A shapefile either consists of points, polylines or polygons, but not more than one.

See this article for more: Shapefiles