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Conclusion: (i) Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration in the question does not exist too. (ii) The symbolic result produced by the Integrate command with the PrincipalValue->True option is not correct.

Conclusion: Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration in the question does not exist too.

Conclusion: (i) Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration in the question does not exist too. (ii) The symbolic result produced by the Integrate command with the PrincipalValue->True option is not correct.

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en[s_]den[s_] = 1 + s/(s^2 - 1)^2;
f[s_] = Exp[I s]/den[s] - Exp[I*s];
integ = Sqrt[2 Pi] Limit[FourierTransform[1/den[s] - 1, s, k], k -> 1];
integ // N
(*-0.998307 - 0.313707 I*)

Conclusion: Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration in the question does not exist too.

en[s_] = 1 + s/(s^2 - 1)^2;
f[s_] = Exp[I s]/den[s] - Exp[I*s];
integ = Sqrt[2 Pi] Limit[FourierTransform[1/den[s] - 1, s, k], k -> 1];
integ // N
(*-0.998307 - 0.313707 I*)

Conclusion: Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration does not exist too.

den[s_] = 1 + s/(s^2 - 1)^2;
f[s_] = Exp[I s]/den[s] - Exp[I*s];
integ = Sqrt[2 Pi] Limit[FourierTransform[1/den[s] - 1, s, k], k -> 1];
integ // N
(*-0.998307 - 0.313707 I*)

Conclusion: Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration in the question does not exist too.

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Following the advice by @b3m2a1, I present my answer. Let us substract Exp[I*s] from the integrand. Then

a = Integrate[Exp[I*s]/(1 + s/(s^2 - 1)^2) - Exp[I*s], {s, -Infinity, Infinity},
PrincipalValue -> True] // ToRadicals;
N[a,30]
(*-1.41400018802696985500835186883 + 0.19227531358741820427339218093 I*)

is in discordance with

NIntegrate[Exp[I*s]/(1 + s/(s^2 - 1)^2) - Exp[I*s],{s, -Infinity,
-(1/(2 Sqrt[3/( 4 + (155/2 - (3 Sqrt[849])/2)^(1/3) + (1/2 (155 + 3 Sqrt[849]))^(1/3))])) - 
 1/2 \[Sqrt](8/3 - 1/3 (155/2 - (3 Sqrt[849])/2)^(1/3) - 
 1/3 (1/2 (155 + 3 Sqrt[849]))^(1/3) + 
 2 Sqrt[3/(      4 + (155/2 - (3 Sqrt[849])/2)^(
   1/3) + (1/2 (155 + 3 Sqrt[849]))^(1/3))]),
-(1/(2 Sqrt[3/(   4 + (155/2 - (3 Sqrt[849])/2)^(1/3) + (1/2 (155 + 3 Sqrt[849]))^(
1/3))])) + 1/2 \[Sqrt](8/3 - 1/3 (155/2 - (3 Sqrt[849])/2)^(1/3) - 
 1/3 (1/2 (155 + 3 Sqrt[849]))^(1/3) + 
 2 Sqrt[3/(4 + (155/2 - (3 Sqrt[849])/2)^(
   1/3) + (1/2 (155 + 3 Sqrt[849]))^(1/3))]),Infinity},
Method -> "PrincipalValue", WorkingPrecision -> 40, AccuracyGoal -> 7, PrecisionGoal -> 7]
(*-0.9983073697207267431817361195580732660872 - .313706787296245329692113892146598026371 I*)

Therefore, at least one of the above results in not correct. However, the latter result is in accordance with the modified approach by @b.gates.you.know.what :

en[s_] = 1 + s/(s^2 - 1)^2;
f[s_] = Exp[I s]/den[s] - Exp[I*s];
integ = Sqrt[2 Pi] Limit[FourierTransform[1/den[s] - 1, s, k], k -> 1];
integ // N
(*-0.998307 - 0.313707 I*)

The above result can be grounded by the calculation with the residues (Here the Jordan's lemma is applicable: pay your attention to -Exp[I*s] in the definition of f[s_].).

Conclusion: Because $$PV\int_{-\infty}^\infty \exp(is)\,ds $$ does not exist, the principal value of the integral under consideration does not exist too.