Rational homology sphere that is not Seifert manifold

By Thurston, all but finitely many $(p,q)$-surgeries on a hyperbolic knot in $S^3$ result in hyperbolic rational homology spheres for $p\neq 0$. In particular there are infinitely many integral homology spheres among them.

If you glue opposite faces of a dedecahedron with a twist of $\frac\pi5$, you obtain the classical Poincaré sphere. It is an integral homology sphere, and a Seifert manifold, see BS' comment below.

But if you do the gluing with a twist of $\frac{3\pi}5$, then you obtain the Seifert-Weber manifold, which is a rational homology sphere with a hyperbolic structure, so definitely not a Seifert manifold. It has $H_1\cong(\mathbb Z/5)^3$, see Neil Hoffman's comment. I do not know what knot would give this manifold. But you can deduce many of its properties by drawing a picture of a dodecahedron with the appropriate corners, edges and so on identified.