Humming and buzzing in an audio system is usually caused by ground loop. There could be exception, like the broken circuit or a damaged microphone capsule. But ground loop explains most of the buzzing cases.
Ground loop causes buzzing!
Having the ground is always the starting point when setting up an audio system. It is essential for safety reason and for noise elimination whenever you plug in sound equipment. But ground loop means the opposite. Basically what happens is that two devices are fighting to decide which one is the ground, specifically two AC devices have their input and output connected together. Such as a powered amp is being used as a mixer output and this two device are power by different wall outlet. Then the horrible high pitch noise is most likely to follow up, it does not matter whether you move the equipment. In this case, if you are trying to get rid of static with noise reduction process in software, you are going to get washout signal. So the post-production just does not help at all. If you can not fix the ground loop, the buzzing will just be there.
How to fix?
On the top of all troubleshoot, you had better to plug everything into the same power strip or wall outlet. One notice here, though almost all the sockets in your house should be properly ground, you still can check if the buzzing comes from socket by investing an outlet tester. Or simply plug all the gears into another socket.
If there are limited sockets or the buzzing still can be heard, we are going to show the party piece - a ground loop isolator. It will break the ground connection between the I/O of two devices that are powered by separate wall outlet. It works as a DI box in a more professional setup where balance and unbalance audio will need to be considered. But what we are talking about is consumer level of audio, which means the input and output are both unbalance, then a simple ground loop isolator will do the job. It grounds out electricity and allows your audio signal to be crisp and clean. It is also easy to add the isolator to the whole setup, take the audio system of a computer and speaker as example, just put it between that two AC powered devices.
So far, the buzzing in most cases should have been solved. If not yet, other troubleshooting you may also want to try out and see if it is a product issue. Use better quality audio cables. They are expensive but they work. Placing audio equipment far from high current power supplies and cables.