Due to the diversity of prospective antenna designs and the wide spectrum of frequencies you described above, we would need more information about what type of installation you can accommodate in a base station environment. Also, I assume that you are not transmitting with these antennas and therefore do not have the instruments needed to properly tune antennas for a given band. You don't mention whether or not you are a licensed Ham. Some questions that need to be addressed:
At home, can you install an outdoor scanner antenna or do restrictions require you to "hide" it in some way?
Scanners cover quite a wide range of VHF/UHF frequencies. While building your own is an enjoyable project (it's what I enjoy most about the radio hobby) at these frequencies there are many factory built antennas which are quite reasonable in price and work well. It is likely that you would need a multi-band design of some type to offer the broad coverage that a scanner requires.
"Feeding" these antennas (routing of the coaxial cable) might also have a large influence on your installation.
Is this handheld radio just a receiver? Is it general coverage (capable of frequencies below 30 MHz) or just VHF/UHF? If it is general coverage, does it have SSB mode? Without SSB you won't hear much amateur radio activity.
For your mobile antennas in the VHF range and above it makes a lot more sense to use a manufactured antenna in my opinion. For HF frequencies (below 30 MHz) then something like a dipole design would allow quick deployment and tear-down while on the road.
Antennas are 100% frequency dependent, one size/design doesn't fit all.