There are several decent candidates (including the bounds on the dimension of the singular locus as mentioned above by Francesco Polizzi).  Here are some others that are somewhat common.

**1.**  *Seminormal and Weakly Normal*.  A reduced scheme $X$ is *seminormal* (resp. *weakly normal*) if for every finite map $f : Y \to X$ satisfying:
  (a)  $Y$ is reduced.
  (b)  $f$ induces a bijection on points (closed or not)
  (c)  $f$ induces isomorphisms of residue fields at each point) (respectively induces a purely inseparable extension of residue fields)

Those 3 conditions is automatically an isomorphism.  For example, a node is seminormal but a cusp is not.  Three lines through the origin in $\mathbb{A}^2$ are not seminormal.  Three coordinate axes through the origin in $\mathbb{A}^3$ is seminormal.  If you don't want to change the points of your variety, seminormalization will make it as nice as you can without messing with those points.

**2.**  *G1 and S2*.  Normality is the same as being regular in codimension 1 (R2) and satisfying Serre's second condition (S2).  A weaker version of this is G1 (Gorenstein in codimension 1) and S2.  The advantage of this is that the canonical module $\omega$ still acts like a divisor on a normal scheme.  The cusp and the node are G1 + S2, but three coordinate lines in $\mathbb{A}^3$ is not.

**3.**  *Seminormal + G1 + S2* (you may as well include equidimensional too).  Combine the best of both worlds...