With the changes to when Ignition effects can be activated, I figured I would do a post on each type of monster effect and when they can be activated.

Ignition Effects:  These effects can only be used during your turn during an open game state in Main Phase 1 or 2.  A list of monsters with Ignition Effects is below.  Please note, your opponent will be able to respond to the summon of these monster's BEFORE they can activate their effects.

Dark Armed Dragon
Judgment Dragon (destruction effect)
Card Trooper (mill effect)
Inzektor Monsters (the effect to equip an Inzektor from hand or grave)
Rescue Rabbit
Black Luster Soldier-Envoy of the Beginning (banish effect)
Red-Eyes Darkness Metal Dragon
Number 17: Leviathan Dragon
Leviair the Sea Dragon
Wind-Up Carrier Zenmaity (summon effect)
Wind-Up Arsenal Zenmaioh
Adreus, Keeper of Armageddon
Wind-Up Hunter
Wind-Up Shark (effect to change levels)
Wind-Up Rat
Lyla, Lightsworn Sorceress (destruction effect)
Lumina, Lightsworn Summoner (destruction effect)
Scrap Dragon
Black Rose Dragon (effect to change an opposing monsters battle position)
Brionac, Dragon of the Ice Barrier
Lonefire Blossom
Spore
Glow-Up Bulb
Blackwing-Gale the Whirlwind
Blackwing-Sirocco the Dawn
Blackwing-Vayu the Emblem of Honor
Test Tiger
Number 20: Giga-Brilliant
Number C39: Utopia Ray
CardCar D
Dark Grepher
Dark End Dragon
XX-Saber Faultroll
Infernity Necromancer (special summon effect)
Infernity Beetle
The Agent of Creation Venus
Master Hyperion

Trigger Effects:  These effects are triggered by something happening, ex. being summoned, destroyed and sent to the graveyard, discarded by a card effect, another card being summoned etc.  If your opponent responds to the summon to destroy the monster, these monster's will still get their effects.  If the summon is negated and the monster's effect is triggered by a summon, then that monster's effect will not activate.  List of monster's with trigger effects is below

Elemental Hero Stratos
Tour Guide from the Underworld
Sangan
Mystic Tomato
Gladiator Beast tagging
Lightsworn Mill
Inzektor Centipede (effect to search when it loses an equip)
Inzektor Dragonfly (effect to special summon when it loses an equip)
T.G. Hyper Librarian
Wind-Up Zenmaines (effect to destroy a card)
T.G. Monsters
XX-Saber Darksoul
XX-Saber Emmersblade
XX -Saber Hyunlei
X-Saber Airbellum
Spirit Reaper (discard effect)
Marshmallon (damage effect)
Blackwing-Shura the Blue Flame
All Monarch's
Elemental Hero The Shining
Elemental Hero Absolute Zero
Elemental Hero Nova Master
Wind-Up Magician
Wind-Up Shark (effect so special summon itself from hand)
T.G. Wonder Magician
Number 39: Utopia
Blackwing-Blizzard the Far North
Infernity Archfiend
Arcanite Magician (effect to gain spell counters)
Gladiator Beast Gyzarus
The Agent of Mystery-Earth
Gorz, the Emissary of Darkness
Tragoedia
Battle Fader
Black Rose Dragon (destruction effect)
All Dark World effects

Quick Effects:  These effects are either A) Monster effects that negation activations/effects of other cards, or B) Cards that say that they can be activated during your opponents turn.  Quick effects are spell speed 2 and therefor can be used in response to summons.  List of monsters with Quick Effects are as follows. 

Maxx "C"
Effect Veiler
Skull Meister
Thunder King Rai-Oh (effect to negate a summon)
Gladiator Beast Heraklinos
Stardust Dragon
Wind-Up Rabbit
Evolzar Laggia
Evolzar Dolkka
Legendary Six Samurai Shi-En
Herald of Orange Light
D.D. Crow
Light and Darkness Dragon
Strike Ninja


Continuous Effects:  These effects do not activate.  They are just applied while the monster is on the field.  They cannot be negated by Shadow-Imprisoning Mirror, Light-Imprisoning Mirror, Divine Wrath, Evolzar Dolkka, Herald of Orange Light. etc.  List of such monsters below

Beast King Barbaros
Spirit Reaper (effect to not be destroyed by battle)
Gellenduo
Marshmallon (effect to not be destroyed by battle)
Arcana Force 0-The Fool
Tragoedia (effect determining it's attack power)
Machina Fortress
Blackwing Armor Master (effect to not be destroyed by battle)
All monsters with Piercing
Aurkus, Lightsworn Druid


I hope this helps people.  Any specific monsters you have questions about please comment below and I will be glad to answer.