It's easy to make unstable systems. There is an old joke that when you try to make an amplifier, you end up with an oscillator, and when you try to make an oscillator, you end up with an amplifier.
The lesson of this joke is that many systems, that you know about, are stable in theory (amps are stable in the linear sense and oscillators are stable in the nonlinear sense), but when implementing them, these real systems have parasitic effects that make the system different than you think. So, start building real circuits and you will find these unstable circuits and wish you never asked the question! (joking, hahaha
)
OK, what about real circuits that are unstable without parasitics? ...
A typical opamp based amplifier circuit driving a capacitive load with too much capacitance will be unstable.
Or, take any first order system, add a delay to the output and then use proportional feedback from the output to the input. As you increase the gain of the proportional feedback, it will eventually go unstable.
Similarly, proportional feedback around a second order system will go unstable if the gain is too high.
Often, proportional-integral feedback is used in the above cases to try and improve control and have good stability margin. Still, if the P-gain and I-gain are not set correctly, instability is easily obtained.