Belatedly, but perhaps of some interest, after some intermittent thought: for non-archimedean $k$ of characteristic $0$ (maybe un-necessarily), non-archimedean, let $A$ be the $n$-by-$n$ matrices, $A^{\le r}$ the matrices of rank $\le r$, $A^{\ge r}$ those of rank at least $r$, and $A^r$ those of rank exactly $r$. Let $G=GL_n(k)$ and $G^1=\{g\in G:|\det g|=1\}$. Let $K=GL_n(\mathfrak o)$, where $\mathfrak o$ is the local integer ring.

We can ask for left-and-right $G^1$-invariant distributions supported on $A^{\le r}$ with $r<n$. Interestingly, $K\times G^1$ is transitive on $A^r$ under the action $(A,B)(x)=A^{-1}xB$. Thus, by uniqueness of invariant distributions, the restriction to $A^{\ge r}$ of a $K\times G^1$-invariant, or $G^1\times K$-invariant distribution supported on $A^\le r$, is unique up to scalar multiples. Indeed, a $G^1\times G^1$-invariant distribution is unique.

In particular, for Schwartz function $f$, the following three integrals give the same outcome up to constants:
$$
\int_K \int_{r\times n} f(k\cdot \pmatrix{0_{n-r} & 0 \cr x_{21} & x_{22}})\;dx_{21}\,dx_{22}\,dk $$
$$
\int_K \int_{n \times r} f(\pmatrix{0_{n-r}&x_{12}\cr 0 & x_{22}}\cdot k)\;dx_{12}\,dx_{22}\,dk
$$
$$
\int_{G^1\times G^1/\Theta} f(g^{-1}\cdot \pmatrix{0_{n-r} & 0 \cr 0 & 1_r}\cdot g')\;dg\,dg'
$$
where $\Theta$ is the isotropy group of $\pmatrix{0_{n-r} & 0 \cr 0 & 1_r}$. Miraculously, $\Theta$ is unimodular, so there is indeed an invariant measure on that quotient.

In particular, the $G\times G$-equivariance is determined by $r$, from those integral formulas: the distribution $v_r$ attached to $r<n$ is homogeneous of degree $r$, in the sense that $v_r(x\to f(A^{-1}xB))=|\det A|^r \cdot |\det B|^r\cdot v_r(f)$: the left equivariance follows from the second integral expression, the right equivariance from the first integral expression.

Fourier transform sends degree-$r$-homogeneous distributions to degree $n-r$. For $n=1$, there is no room to have a $G^1\times G^1$-invariant distribution with support on singular matrices whose Fourier transform is supported likewise. But for $n\ge 2$, there are unique candidates for those invariant distributions, with a unique (up to scalars) distribution for each $r$ in the range $0<r<n$, and Fourier transform replacing $r$ by $n-r$ (so rank $0$ support has Fourier transform consisting of constants, with non-singular support).

The meromorphic family of equivariant distributions $u_s=|\det x|^{s-n}$ has a meromorphic continuation, and the $s$-th one is homogeneous of degree $s$. The residue at $s=r$ for $0<r<n$ is a constant multiple of $v_r$...