The ultimate sparseness occurs when $Ax^*(\lambda)=0$, which is the case when $2\lambda d<\|A\|_{\ell^2\to\ell^1}$. Here $d$ is the Euclidean distance from $b$ to the kernel of $A$. Indeed, the minimizer $x^*$ will be the projection of $b$ onto $\ker A$.

The ultimate sparseness occurs when $Ax^*(\lambda)=0$, which is the case when the minimizer $x^*$ is the projection of $b$ onto $\ker A$. For this to happen, $\lambda$ must be small enough so that the restriction of $A$ to the orthogonal complement of its kernel is bounded from below by a constant greater than $2\lambda \mathrm{dist}(b,\ker A)$. Here the lower bound for operator is understood in the $\ell^2\to\ell^1$ norm.

Perhaps this is a stupid example: $n=2$, $m=1$, $Ax=x_1-x_2$ and $b=(3,1)$. For any $\lambda>0$ the minimizer $x^*$ is on the line segment between $b$ and $(2,2)$, hence all coordinates of $x^*$ are nonzero. So, any $\lambda>0$ is a "minimizer" no matter how large or small.

The ultimate sparseness occurs when $Ax^*(\lambda)=0$, which is the case when $2\lambda d<\|A\|_{\ell^2\to\ell^1}$. Here $d$ is the Euclidean distance from $b$ to the kernel of $A$. Indeed, the minimizer $x^*$ will be the projection of $b$ onto $\ker A$.