What types of antennae do aircraft have and what are their functions? Examples are HF-, VOR- and SATCOM-Antennae.
Like everything in aviation this depends on how the plane is configured and what equipment it contains. This question is pretty broad and there are even some non aviation related antennae on modern commercial planes for things like onboard internet but lets cover the aviation related ones. AOPA provides a great document on it here.
Any plane equipped with GPS will have a GPS unit somewhere on the airframe (usually on the top)
There are generally 2 radio antennas for the communication radios. Traditionally one is mounted on the top of the plane and the other on the bottom.
On smaller GA planes they generally look like this
If your plane is equipped with VOR receiving capability you will have a VOR antenna as well which is generally mounted on the tail.
You will also have a UHF antenna for your transponder and DME, its located on the bottom of the plane
If equipped with an ELT you will also have an ELT antenna, generally on top.
It should be noted this is the general setup of most smaller GA planes. Large commercial planes have the same general equipment but often have greater redundancy and/or more units than your typical GA plane. Some airframes also have antennas for systems that are being phased out like Loran based navigation (which I think has been all but shut down now).
The Boeing 787 has just about every kind of antenna I can think of. Here's a chart of all their locations and purposes.
From Boeing 787 Systems Manual