*For future references. Feel free to edit to include new cases, or any improvements.*

As for 2015, the standard conjectures on algebraic cycles is unconditionally (at lest) known for $X$:

**Lefschetz standard conjecture** (Grothendieck conjectures $A(X)$ and $B(X)$)

- a curve (trivial).
- a surface with $H^1(X)=2\cdot\mathrm{Pic}^0(X)$ (Grothendieck).
- an abelian variety (Liebermann).
- a generalized flag manifold $G/P$ (Schubert).
- a smooth varieties which is complete intersections in some projective space (trivial).
- a Grassmannian (Liebermann?).
- for which $H^*(X)$ is isomorphic to the Chow ring $A^*(X)$.
- a smooth projective moduli space of sheaves on rational Poisson surfaces.
- a uniruled threefold (Arapura).
- a unirational fourfold (Arapura).
- the moduli space of stable vector bundles over a smooth projective curve (Arapura).
- the Hilbert scheme $S^{[n]}$ of a smooth projective surface (Arapura).
- a smooth projective variety of $K3^{[n]}$-type (Charles and Markman).

**Weak Lefschetz standard conjecture** (Grothendieck conjecture $C(X)$)

- all of the above, since $B(X) \Rightarrow C(X)$ (Grothendieck).
- defined by equations with coefficients in a finite field (Katz and Messing).

**Hodge standard conjecture** (Grothendieck conjecture $Hdg(X)$)

- in characteritic $0$ (Hodge).
- a surface (Segre, Grothendieck).

*Main references:*

Alexander Grothendieck, "Standard Conjectures on Algebraic Cycles" (1969).

Steven Kleiman, "The standard conjectures" (1994).

Francois Charles, Eyal Markman, The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of $K3$ surfaces (2011)

*References for proofs:*

Beniamino Segre, Intorno ad teorema di Hodge sulla teoria della base per le curve di una sperficie algebraica (1937)

Alexander Grothendieck, "Sur une note de Mattuck-Tate" (1958)

D. I. Lieberman, "Higher Picard Varieties" (1968)

N. Katz and W. Messing, "Some consequences of the Riemann hypothesis for varieties over finite fields" (1973)

Donu Arapura, Motivation for Hodge cycles (2005)

The standard conjecturesby S. Kleiman, in the Proceedings of the AMS Summer Conference on Motives (this volume). I don't think there has been any notable progress since then. – abx Jul 15 '14 at 6:25reallyunderstood what motives are, as long as we have not answered the standard conjectures on algebraic cycles. – jmc Apr 7 '15 at 12:18