In all the literature I have read, etale cohomology is defined for smooth varieties. Suppose $X/\mathbb{Q}$ is a singular variety over $\mathbb{Q}$, does there exist etale cohomology $H\_{\text{et}}^\*(X\_{\bar{\mathbb{Q}}},\mathbb{Q}\_{\ell})$ defined for it? If $H\_{\text{et}}^\*(X\_{\bar{\mathbb{Q}}},\mathbb{Q}\_{\ell})$ is defined, is it still a representation of the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$? Does the characteristic polynomial of the geometric Frobenius still counts the rational points of $X$? 



Etale cohomology of singular varieties?
Just to answer the last question, the key statement is that for a proper variety over $\mathbb F\_q$, the number of $\mathbb F\_{q^n}$ points is equal to the alternating sum of the traces of $\operatorname{Frob}\_{q^n} = \operatorname{Frob}\_q^n$ on the etale cohomology. (Or for a general variety, if you take etale cohomology with compact supports). 


If you have a variety $X$ over $\mathbb Q$, take a model over $\mathbb Z$, and look at the $\mathbb F\_{p^n}$-points, then you can get this statement in terms of the trace of the Frobenius element $\operatorname{Gal}(\overline{\mathbb Q}|\mathbb Q)$ acting on $H^\*\_{et}(X\_{\overline{\mathbb Q}} , \mathbb Q\_\ell)$ if you can show that the cohomology is ``the same" in characteristic zero and characteristic $p$. More precisely, you want to show that the cohomology sheaves on $\operatorname {Spec} \mathbb Z$ are lisse. One case you can do this is when the integral model you have chosen is smooth and proper over $\mathbb Z$. This is where a smoothness assumption is helpful.


However, if you compute directly the etale cohomology in characteristic $p$, the smoothness is not needed for the Lefschetz fixed point formula.


Also, if you have a variety over $\mathbb Q$ and are comparing its reductions to different primes $p$, for any given integral model this lisseness assumption will hold for all but finitely many primes $p$, so if you are willing to throw away finitely many primes you don't need any additional assumptions.. 


