This plot shows that the fundamental mode has one main lobe in broadside, the TM20 mode has two lobes symmetrically around broadside with a radiation null in broadside and the TM30 mode has three lobes: one in broadside and two symmetrically next to the broadside lobe.
These are for a rectangular patch only and differently shaped patch antennas exhibit other mode behavior and associated radiation patterns. Another excellent example is the round patch antenna shown in Figure 24: