Thread a list of replacement rules into a list

You can useMapThread:

list = {-1 - ϵ + 1/2 ϵ^2 x''[0],  1 - ϵ + 1/2 ϵ^ 2 x''[0]};
rules = {x''[0] -> -1, x''[0] -> 1};

MapThread[ReplaceAll,  {list, rules}]

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

You can also use Inner as follows:

Inner[ReplaceAll, list, rules, List]

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

How to do it with Thread:

If you use Thread you need to wrap ReplaceAll[...] with Unevaluated to prevent ReplaceAll being evaluated before Thread gets to do its job:

Thread[Unevaluated[ReplaceAll[list, rules]]]

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

Another way to use Thread:

ReplaceAll @@@ Thread[{list, rules}]

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

Least Mathematica way:

My candidate for doing this in the least Mathematicaesq way in Mathematica is:

For[results = {}; i = 1, i <= 2, i++, AppendTo[results, list[[i]] /. rules[[i]]]]
results

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

Halloween specials:

☺ = # /. #2 & @@@ ({##}\[Transpose]) &;
☺[list, rules]

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

☺☺ = {# /. #3, #2 /. #4} & @@ (## & @@@ {##}) &;
☺☺[list, rules]

{-1 - ϵ - ϵ^2/2, 1 - ϵ + ϵ^2/2}

"The least Mathematica way possible"? Something like this?

lst = {-1 - ϵ + 1/2 ϵ2 x''[0], 1 - ϵ + 1/2 ϵ2 x''[0]};

rules = {x''[0] -> -1, x''[0] -> 1};

Table[ lst[[i]] /. rules[[i]], {i, Length@rules}]

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