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Every constructible sheaf of $\mathbb{Q}-$modules over $\mathbb{R}$ is the direct sum of indecomposable sheaves, which are either sheaves with stalk $\mathbb{Q}$ at a point or constant sheaves with stalk $\mathbb{Q}$ on an open interval extended by lower ! or lower * at each endpoint. What happens if we remove the 'constructible' hypothesis? Is there a similar classification?

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Are you looking for something that says, roughly, that the category of sheaves over $\mathbb R$ is a "limit" of the categories of constructible sheaves with respect to a given stratification? Here, the "limit" should be taken with respect to the poset of stratifications of $\mathbb R$. –  André Henriques Mar 25 '11 at 14:12

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