diff --git a/Project.toml b/Project.toml
index 2d1808607..6bbf26462 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "LinearSolve"
 uuid = "7ed4a6bd-45f5-4d41-b270-4a48e9bafcae"
 authors = ["SciML"]
-version = "2.11.1"
+version = "2.12.0"
 
 [deps]
 ArrayInterface = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
diff --git a/src/LinearSolve.jl b/src/LinearSolve.jl
index e240a790b..661a0a859 100644
--- a/src/LinearSolve.jl
+++ b/src/LinearSolve.jl
@@ -143,6 +143,46 @@ end
     include("factorization_sparse.jl")
 end
 
+# Solver Specific Traits
+## Needs Square Matrix
+"""
+    needs_square_A(alg)
+
+Returns `true` if the algorithm requires a square matrix.
+
+Note that this checks if the implementation of the algorithm needs a square matrix by
+trying to solve an underdetermined system. It is recommended to add a dispatch to this
+function for custom algorithms!
+"""
+needs_square_A(::Nothing) = false  # Linear Solve automatically will use a correct alg!
+function needs_square_A(alg::SciMLLinearSolveAlgorithm)
+    try
+        A = [1.0 2.0;
+            3.0 4.0;
+            5.0 6.0]
+        b = ones(Float64, 3)
+        solve(LinearProblem(A, b), alg)
+        return false
+    catch err
+        return true
+    end
+end
+for alg in (:QRFactorization, :FastQRFactorization, :NormalCholeskyFactorization,
+    :NormalBunchKaufmanFactorization)
+    @eval needs_square_A(::$(alg)) = false
+end
+for kralg in (Krylov.lsmr!, Krylov.craigmr!)
+    @eval needs_square_A(::KrylovJL{$(typeof(kralg))}) = false
+end
+for alg in (:LUFactorization, :FastLUFactorization, :SVDFactorization,
+    :GenericFactorization, :GenericLUFactorization, :SimpleLUFactorization,
+    :RFLUFactorization, :UMFPACKFactorization, :KLUFactorization, :SparspakFactorization,
+    :DiagonalFactorization, :CholeskyFactorization, :BunchKaufmanFactorization,
+    :CHOLMODFactorization, :LDLtFactorization, :AppleAccelerateLUFactorization,
+    :MKLLUFactorization, :MetalLUFactorization)
+    @eval needs_square_A(::$(alg)) = true
+end
+
 const IS_OPENBLAS = Ref(true)
 isopenblas() = IS_OPENBLAS[]
 
diff --git a/test/nonsquare.jl b/test/nonsquare.jl
index c678c17b3..e0e817b9e 100644
--- a/test/nonsquare.jl
+++ b/test/nonsquare.jl
@@ -8,7 +8,11 @@ b = rand(m)
 prob = LinearProblem(A, b)
 res = A \ b
 @test solve(prob).u ≈ res
+@test !LinearSolve.needs_square_A(QRFactorization())
 @test solve(prob, QRFactorization()) ≈ res
+@test !LinearSolve.needs_square_A(FastQRFactorization())
+@test solve(prob, FastQRFactorization()) ≈ res
+@test !LinearSolve.needs_square_A(KrylovJL_LSMR())
 @test solve(prob, KrylovJL_LSMR()) ≈ res
 
 A = sprand(m, n, 0.5)
@@ -23,6 +27,7 @@ A = sprand(n, m, 0.5)
 b = rand(n)
 prob = LinearProblem(A, b)
 res = Matrix(A) \ b
+@test !LinearSolve.needs_square_A(KrylovJL_CRAIGMR())
 @test solve(prob, KrylovJL_CRAIGMR()) ≈ res
 
 A = sprandn(1000, 100, 0.1)
@@ -35,7 +40,9 @@ A = randn(1000, 100)
 b = randn(1000)
 @test isapprox(solve(LinearProblem(A, b)).u, Symmetric(A' * A) \ (A' * b))
 solve(LinearProblem(A, b)).u;
+@test !LinearSolve.needs_square_A(NormalCholeskyFactorization())
 solve(LinearProblem(A, b), (LinearSolve.NormalCholeskyFactorization())).u;
+@test !LinearSolve.needs_square_A(NormalBunchKaufmanFactorization())
 solve(LinearProblem(A, b), (LinearSolve.NormalBunchKaufmanFactorization())).u;
 solve(LinearProblem(A, b),
     assumptions = (OperatorAssumptions(false;