Well, the number of applications have fallen off due to power 2.0 trying to kill practical logic, but uses include(d):
-Chaindrives: used for perma-jumping
-Logic inhibitors: staged jump inhibitors create a continuous inhibitor signal which effectively doubles it's effectiveness.
-Staged turret systems: By connecting your turrets to a logic system you can create firing patterns to do all sorts of things like fire ion ahead of block weapons, reduce power capacity requirements, etc.
- These same systems also doubled for turning on/off fire-on-target turrets to prevent friendly fire.
-Staged swarmers: staging swarmers is great for killing drone fleets and acting as distractors.
-Logic guns: less accurate than traditional hull mounted weapon systems, but can be much more powerful through staging, and can also do unfocused fire in a way that staged turrets can't
-Auto-defence systems for killing boarding astronauts.
-Hanger logic for a recon shuttle and/or escape pod(s)
-Laminar or wave shield activator (to reduce lag when not in use)
-Raildocker doors: used to keep people from force docking you with boarding ships or tracker tags.
-Detector Turrets: Fire-at-any turret used as a sensor system to activate certain kinds of systems such as jump inhibitors, or just to alert you of nearby enemies with something like a blinking light.
I'm sure I'm missing a few, but these are probably the most common.
Some more uses just for fun including some things that got patched pre-power update:
- AMS cycling: keeps AMS from getting too distracted by distraction missiles.
- Warhead hammers: use of area trigger and rails to swing or push warheads with enough force to do a lot of damage regardless of the actual speed and trajectory of your torpedo.
- Drone/torpedo/chaff launchers.
- Logic Jammers: Set on a cycle, multiple jammers could be combined with a logic system to reset jamming quickly after getting scanned.
- Zero-Point Injectors: Used to exploit power and shield support systems.
- Pulse Drives: Like chain drives, but uses a single warp core and logic multiplexing.
- Curtain AMS: auto fire AMS waffles.
- Speed Maneuvering Systems: Use of dock/undock with turning rails to make ships that could do weird hair-pin turns.