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..b0511080a 100644
--- a/src/LinearSolve.jl
+++ b/src/LinearSolve.jl
@@ -62,6 +62,9 @@ needs_concrete_A(alg::AbstractKrylovSubspaceMethod) = false
 needs_concrete_A(alg::AbstractSolveFunction) = false
 
 # Util
+is_underdetermined(x) = false
+is_underdetermined(A::AbstractMatrix) = size(A, 1) < size(A, 2)
+is_underdetermined(A::AbstractSciMLOperator) = size(A, 1) < size(A, 2)
 
 _isidentity_struct(A) = false
 _isidentity_struct(λ::Number) = isone(λ)
@@ -96,6 +99,7 @@ EnumX.@enumx DefaultAlgorithmChoice begin
     NormalCholeskyFactorization
     AppleAccelerateLUFactorization
     MKLLUFactorization
+    QRFactorizationPivoted
 end
 
 struct DefaultLinearSolver <: SciMLLinearSolveAlgorithm
@@ -143,6 +147,31 @@ 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.
+"""
+needs_square_A(::Nothing) = false  # Linear Solve automatically will use a correct alg!
+needs_square_A(alg::SciMLLinearSolveAlgorithm) = true
+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/src/default.jl b/src/default.jl
index 509a14571..82166a247 100644
--- a/src/default.jl
+++ b/src/default.jl
@@ -1,6 +1,6 @@
 needs_concrete_A(alg::DefaultLinearSolver) = true
 mutable struct DefaultLinearSolverInit{T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12,
-    T13, T14, T15, T16, T17, T18}
+    T13, T14, T15, T16, T17, T18, T19}
     LUFactorization::T1
     QRFactorization::T2
     DiagonalFactorization::T3
@@ -19,6 +19,7 @@ mutable struct DefaultLinearSolverInit{T1, T2, T3, T4, T5, T6, T7, T8, T9, T10,
     NormalCholeskyFactorization::T16
     AppleAccelerateLUFactorization::T17
     MKLLUFactorization::T18
+    QRFactorizationPivoted::T19
 end
 
 # Legacy fallback
@@ -168,8 +169,8 @@ function defaultalg(A, b, assump::OperatorAssumptions)
                        (A === nothing ? eltype(b) <: Union{Float32, Float64} :
                         eltype(A) <: Union{Float32, Float64})
                     DefaultAlgorithmChoice.RFLUFactorization
-                    #elseif A === nothing || A isa Matrix
-                    #    alg = FastLUFactorization()
+                #elseif A === nothing || A isa Matrix
+                #    alg = FastLUFactorization()
                 elseif usemkl && (A === nothing ? eltype(b) <: Union{Float32, Float64} :
                         eltype(A) <: Union{Float32, Float64})
                     DefaultAlgorithmChoice.MKLLUFactorization
@@ -199,9 +200,19 @@ function defaultalg(A, b, assump::OperatorAssumptions)
     elseif assump.condition === OperatorCondition.WellConditioned
         DefaultAlgorithmChoice.NormalCholeskyFactorization
     elseif assump.condition === OperatorCondition.IllConditioned
-        DefaultAlgorithmChoice.QRFactorization
+        if is_underdetermined(A)
+            # Underdetermined
+            DefaultAlgorithmChoice.QRFactorizationPivoted
+        else
+            DefaultAlgorithmChoice.QRFactorization
+        end
     elseif assump.condition === OperatorCondition.VeryIllConditioned
-        DefaultAlgorithmChoice.QRFactorization
+        if is_underdetermined(A)
+            # Underdetermined
+            DefaultAlgorithmChoice.QRFactorizationPivoted
+        else
+            DefaultAlgorithmChoice.QRFactorization
+        end
     elseif assump.condition === OperatorCondition.SuperIllConditioned
         DefaultAlgorithmChoice.SVDFactorization
     else
@@ -247,6 +258,12 @@ function algchoice_to_alg(alg::Symbol)
         NormalCholeskyFactorization()
     elseif alg === :AppleAccelerateLUFactorization
         AppleAccelerateLUFactorization()
+    elseif alg === :QRFactorizationPivoted
+        @static if VERSION ≥ v"1.7beta"
+            QRFactorization(ColumnNorm())
+        else
+            QRFactorization(Val(true))
+        end
     else
         error("Algorithm choice symbol $alg not allowed in the default")
     end
@@ -311,6 +328,12 @@ function defaultalg_symbol(::Type{T}) where {T}
 end
 defaultalg_symbol(::Type{<:GenericFactorization{typeof(ldlt!)}}) = :LDLtFactorization
 
+@static if VERSION >= v"1.7"
+    defaultalg_symbol(::Type{<:QRFactorization{ColumnNorm}}) = :QRFactorizationPivoted
+else
+    defaultalg_symbol(::Type{<:QRFactorization{Val{true}}}) = :QRFactorizationPivoted
+end
+
 """
 if alg.alg === DefaultAlgorithmChoice.LUFactorization
     SciMLBase.solve!(cache, LUFactorization(), args...; kwargs...))
diff --git a/src/factorization.jl b/src/factorization.jl
index 7156991fb..2917580a3 100644
--- a/src/factorization.jl
+++ b/src/factorization.jl
@@ -158,6 +158,16 @@ function QRFactorization(inplace = true)
     QRFactorization(pivot, 16, inplace)
 end
 
+@static if VERSION ≥ v"1.7beta"
+    function QRFactorization(pivot::LinearAlgebra.PivotingStrategy, inplace::Bool = true)
+        QRFactorization(pivot, 16, inplace)
+    end
+else
+    function QRFactorization(pivot::Val, inplace::Bool = true)
+        QRFactorization(pivot, 16, inplace)
+    end
+end
+
 function do_factorization(alg::QRFactorization, A, b, u)
     A = convert(AbstractMatrix, A)
     if alg.inplace && !(A isa SparseMatrixCSC) && !(A isa GPUArraysCore.AbstractGPUArray)
diff --git a/test/nonsquare.jl b/test/nonsquare.jl
index c678c17b3..694bf4dfd 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;
@@ -49,3 +56,12 @@ solve(LinearProblem(A, b), (LinearSolve.NormalCholeskyFactorization())).u;
 solve(LinearProblem(A, b),
     assumptions = (OperatorAssumptions(false;
         condition = OperatorCondition.WellConditioned))).u;
+
+# Underdetermined
+m, n = 2, 3
+
+A = rand(m, n)
+b = rand(m)
+prob = LinearProblem(A, b)
+res = A \ b
+@test solve(prob).u ≈ res