Light Mortar is useful now that it has 600 range. The main reason not use to Light Mortar is the ships that can use it well become obsolete after early game. On Centurion, I like two light mortars over one light assault gun due to flux efficiency. It does not have the flux efficiency to use four top-of-the-line light weapons, but it can handle three, and two light mortars are cheaper than one light assault gun.
Dual light MG costs too much OP. It might have use for Lashers later, except Lashers are no good later.
Light Needler is now a railgun clone that costs too much OP. It is a classic "don't fix something that was not broken", and now it is sub-par for its cost. It may have use if you absolutely need 700 range light ballistic, and you have light needler blueprint but not railgun.
LRPD laser only costs 30 flux since 0.8. It is a little underpowered and does not excel at anything, but it is still a decent all-purpose beam weapon.
Mining laser is bad because it is too slow. Yes, it is underpowered, but that would be tolerable if it was as quick as a PD laser. As nearly as slow as tactical laser, it activates too late to be of any use. If it was faster, it would be useful for times when player obtains IPDAI hullmod before the high-tech pack.
I occasionally use singleton missiles, especially if I have a few spare OP and nothing better to use.
Assault Chaingun is useless except for destroyers and some cruisers with Safety Override, and those become obsolete before endgame.
HMG is a close-range assault weapon. It is a very niche weapon with or without Safety Override.
Old pre-0.9 heavy needler would be useful on Legion due to flux efficiency, but at its cost, I prefer Heavy AC over it. At least Heavy Needler still has 800 range.
Thumper is a decent early game weapon for shredding weakened opponents. It becomes obsolete eventually after player accumulates better, but it can be useful early when player does not have everything.
Ion pulser seems tricky to use and not as reliable as the more ordinary pulse laser. It may not be a bad weapon per se, but I prefer pulse laser over it, and leave the stunning to Ion Cannon if I am so inclined.
Mining blaster now has the dubious honor of being the default starter weapon of its type (of medium energy) if you have not found any blueprints yet. Flux efficiency is a killer, but if you have nothing else, you use what you have (or avoid those ships that use energy until you find better). Aside from that, it is only optimal on Hyperion.
I like annihilator pod, one of the few missile weapons I do not mind using. Lasts longer than Harpoons and more versatile. I use this on Falcon (P), Dominator, and Onslaught.
Proximity charges are too overpriced for what they do. Let Flash wings use this weapon, not you.
Storm Needler pairs alright with Heavy Mortars on Conquest, and is useful if you have Storm Needlers handy but little else.
Paladin is overshadowed by other options on the two ships that can use it effectively.