The spectral properties of a variety of improved staggered operators are studied in quenched QCD. The systematic dependence of the infrared eigenvalue spectrum on i) improvement in the staggered operator, ii) improvement in the gauge field action, iii) lattice spacing and iv) lattice volume, is analyzed. It is observed that eigenmodes with small eigenvalues and large chirality appear as the level of improvement increases or as one approaches the continuum limit. These eigenmodes can be identified as the ``zero modes'' which contribute to the chirality associated, via the index theorem, with the topology of the background gauge field. This gives evidence that staggered fermions are sensitive to gauge field topology. After successfully identifying these would-be chiral zero modes, the distribution of the remaining non-chiral modes is compared with the predictions of Random Matrix Theory in different topological sectors. Satisfactory agreement is obtained.

10 pages, 11 Figures, Final version