This is probably a dumb question, but I am sort of confused. What direction does a cap discharge when a system is turned off? For example, decoupling caps that go to ground. When I shut a system down, does the current flow to ground or does it flow in the reverse direction toward the supply (since it is now 0). Would putting a resistor in series to ground have any effect?
For example, decoupling caps that go to ground.
Caps block DC, so V+ will want to return to ground through the circuit, as usual, taking the easiest path. Decoupling caps are not usually the issue. It is the bulk caps that act like batteries when the power is disconnected.
When I shut a system down, does the current flow to ground
Current only flows toward lower voltages. If voltage is trapped in the circuit, either because the switch physically disconnected V+, or because the power cord was physically disconnected, the device will continue trying to work, consuming the remaining power. Caps will eventually dissipate their charge if there's nowhere for it to go.
or does it flow in the reverse direction toward the supply (since it is now 0).
If a transformers input is switched, but the output is not, the V+ terminal may be the shortest path back to 0V. However, a rectifier between transformer and circuit negates that, leaving GND the only path back.
Would putting a resistor in series to ground have any effect?
A resistor in parallel is called a "bleeder" and is common when the circuit may inadvertently retain unwanted/dangerous voltages. They are basically "soft" shorts, in case the circuit's GND gets disconnected. The drawback of a bleeder is that it dissipates power whether the circuit is operating or being bled.