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PostWotLKEdit
With the release of Wrath of the Lich King, Incinerate has become the best option for deep Destruction specs, while Shadow Bolt has remained the best option for deep Affliction specs. Shadow Bolt is also the best option for deep Demonology and Affliction / Demonology hybrid specs, as they don't have enough spare talent points to access the Destruction talents which specifically improve Incinerate or fire spells. For hybrid specs, the question is about how many points must be invested in Destruction before Incinerate out damages Shadow Bolt. Shadow Bolt and Incinerate deal fairly similar DPS, and before the effects of talents, the DPS of both spells scales at the same rate with spell power. The tier 1 Destruction talents Bane and Improved Shadow Bolt combine to increase Shadow Bolt's DPS by 26% when maxed out, while maxing out the tier 6 Destruction talent Emberstorm increases Incinerate's DPS by nearly 28%, 4 ranks of Emberstorm increase Incinerate's DPS by slightly under 22%, placing the tipping point firmly at 30 points in Destruction for Demonology / Destruction hybrids. Affliction's tier 6 talent Shadow Mastery increases shadow damage by 15% when maxed out, when combined with Bane, this adds up to a 38% DPS boost for Shadow Bolt, making Shadow Bolt a better option for Affliction / Destruction hybrids (although Affliction / Destruction hybrid builds are not recommended due to lack of synergy between the two talent trees). Obviously, immunities, resistances and absorption can change things. Less obviously, Immolate is generally not a significant factor, as Incinerate's conditional damage does not scale with spell power.
Molten Core in Patch 3.3.0Edit
As of PTR build 10623, Molten Core has been redesigned. Among its' other effects, it gives each Corruption tick a 4%/8%/12% chance per tick to make the next three casts of Incinerate deal 6%/12%/18% more damage and reduce their cast time by 10%/20%/30%. At rank 1, this translates to a roughly 17.8% DPS increase for Incinerate. At rank 2, this translates to a 40% DPS increase for Incinerate. At rank 3, this translates to a roughly 68.6% DPS increase for Incinerate. This makes Incinerate better than Shadow Bolt for deep Demonology and Demonology hybrid specs while the Molten Core buff is active.
PreWotLKEdit
Prior to Wrath of the Lich King, a common debate among Destruction warlocks is which is better, Fire or Shadow? It boils down to simple Mathematics.
First we must ask ourselves "What are we concerned with?" for the topic of this debate. Let's consider sustained DPS in raid situations.
MathEdit
Common BonusesEdit
Common Bonuses are things that both spells receive the same bonus from and add multiplicatively with the rest of the equation.
 Hit Rating
 Haste Rating
That's it! Everything else impacts the equations used to determine damage differently.
Incinerate dataEdit
Deals 479 damage on average Deals 599 damage on average with immolate present 2.5 cast time (talents to 2.25) receives 2.5/3.5 percent of spell damage (talents to .2+2.5/3.5)
 Mob Debuff: Misery (+5%)
 Mob Debuff: Curse of Elements (+10%)
 Mob Debuff: Improved Scorch (+15%)
 Self Buff: Emberstorm (+10%)
 Self Buff: Imp Demonic Sacrifice (15%)
so an equation modeling Incinerate's DPS would look Something like This (where S is spell damage and C is Crit % represented as a number between 1 and 0)
$ ((1+C)(1.05)(1.10)(1.15)(1.10)(1.15)(599+(.2+2.5/3.5)S))/2.25 $
Where (1+C) is the damage increase caused by your crit rate, C. The 1.XX are the total damage multipliers. These multipliers are then each multiplied against the quantity of the average damage plus the talented spell power coefficient of incinerate times the spell power,(599+(.2+2.5/3.5)S), in order to find the average damage of a hit. The average damage is then divided by casting time to find the dps.
Shadow Bolt dataEdit
Deals 572 damage on average 3 cast time (talents to 2.5) receives 3/3.5 percent of spell damage (talents to 1.2*3/3.5)
 Mob Debuff: Curse of Elements (+10%)
 Mob Debuff: Misery (+5%)
 Mob Debuff: Improved Shadow Bolt (+$ C^4+4C^3+6C^2+4C $%)
 Mob Debuff: Shadow Weaving (+10%)
 Self Buff: Imp Demonic Sacrifice (15%)
so an equation modeling Shadow Bolt's DPS would look Something like This (where S is spell damage and C is Crit % represented as a number between 1 and 0)
$ ((1+C)(1+0.2(C^4+4C^3+6C^2+4C)(1.1)(1.05)(1.1)(1.15)(572+(.2+3/3.5)S))/2.5 $
Where (1+C) is the damage increase caused by your crit rate, C. The 1.XX are the total damage multipliers while the multiplier for ISB is (1+0.2(C^4+4C^3+6C^2+4C). These multipliers are then each multiplied against the quantity of the average damage plus the talented spell power coefficient of Shadowbolt times the spell power,(572+(.2+3/3.5)S), in order to find the average damage of a hit. The average damage is then divided by casting time to find the dps.
Spell CompareEdit
We know Shadow Bolt starts out lower and that if it ever passes Incinerate there must exist an intercept thus any value of Spell Damage/ Crit that exceeds the value at which the interception occurs would be the values at which Shadow Bolt is better than Incinerate.
First set the equations equal to each other.
 $ ((1+C)(1+0.2(C^4+4C^3+6C^2+4C)(1.1)(1.05)(1.1)(1.15)(572+(.2+3/3.5)S))/2.5=((1+C)(1.05)(1.10)(1.15)(1.10)(1.15)(599+(.2+2.5/3.5)S))/2.25 $
Then simplify the multipliers except ISB since it includes crit.
 $ ((1+C)(1+0.2(C^4+4C^3+6C^2+4C)(1.461075)(572+(.2+3/3.5)S)/2.5=(1+C)(1.68023625)(599+(.2+2.5/3.5)S)/2.25 $
The 1+C is common among both so it can be eliminated.
 $ (1+0.2(C^4+4C^3+6C^2+4C)(1.461075)(572+(.2+3/3.5)S)/2.5=(1.68023625)(599+(.2+2.5/3.5)S)/2.25 $
I like not having fractions so multiply the speed of SB on both sides to eliminate it, from this we verify that the faster cast rate of incinerate increases it's dps. I also simplified the spell damage coefficients.
 $ (1+0.2(C^4+4C^3+6C^2+4C)(1.461075)(572+1.071S)=(1.68023625)(599+.91743S)(1.11) $
Then just multiply the multipliers against the average damage to find the buffed average damage and the buffed spell damage coefficient.
 $ (1+0.2(C^4+4C^3+6C^2+4C)(835.7349+1.564811325S)=(1117.1722802625+1.711064048549625S) $
(lets assume 35% crit since this is sustained dps and most destro locks can obtain this pretty easily)
Using the result from the last section we have:
 $ (1+0.2(C^4+4C^3+6C^2+4C)(835.7349+1.564811325S)=(1117.1722802625+1.711064048549625S) $
Then substitute the crit in using C=.35.
 $ (1+0.2(0.82149375)(835.7349+1.564811325S)=1117.1722802625+1.711064048549625S $
Simplify the multiplier of ISB.
 $ (1.16429875)(835.7349+1.564811325S)=1117.1722802625+1.711064048549625S $
Multiply ISB throughout the SB equation.
 $ 973.045099401375+1.82190786968334375S=1117.1722802625+1.711064048549625S $
Find the difference of the spell power coefficients times S and set it equal to the difference in buffed average hit.
 $ 0.11084382113371875S=144.127180861125 $
Solve for S.
 $ S=1300.2725761975868411427185876779 $
ConclusionsEdit
Incinerate is better for sustained personal dps given that immolate is applied for you (until you have 1.3k spell damage).
A good candidate for keeping immolate up is the designated support warlock who has 5/5 Shadow Embrace, 3/3 Malediction, on imp duty and/or has the least spell damage.