  This paper studies the singularities of affine Schubert varieties in the
affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of
affine Schubert varieties, we determine the singular loci; and for one class,
we also determine explicitly the tangent spaces at singular points. For a
general affine Schubert variety, we give partial results on the singular locus.
