diff --git a/docs/src/solvers/solvers.md b/docs/src/solvers/solvers.md
index 8b2d54c45..f6c828edb 100644
--- a/docs/src/solvers/solvers.md
+++ b/docs/src/solvers/solvers.md
@@ -82,6 +82,12 @@ use `Krylov_GMRES()`.
 
 ## Full List of Methods
 
+### Polyalgorithms
+
+```@docs
+LinearSolve.DefaultLinearSolver
+```
+
 ### RecursiveFactorization.jl
 
 !!! note
diff --git a/src/LinearSolve.jl b/src/LinearSolve.jl
index af2d94eb4..63a58a0c3 100644
--- a/src/LinearSolve.jl
+++ b/src/LinearSolve.jl
@@ -119,8 +119,23 @@ EnumX.@enumx DefaultAlgorithmChoice begin
     KrylovJL_LSMR
 end
 
+"""
+    DefaultLinearSolver(;safetyfallback=true)
+
+The default linear solver. This is the algorithm chosen when `solve(prob)`
+is called. It's a polyalgorithm that detects the optimal method for a given
+`A, b` and hardware (Intel, AMD, GPU, etc.).
+
+## Keyword Arguments
+
+* `safetyfallback`: determines whether to fallback to a column-pivoted QR factorization
+  when an LU factorization fails. This can be required if `A` is rank-deficient. Defaults
+  to true.
+"""
 struct DefaultLinearSolver <: SciMLLinearSolveAlgorithm
     alg::DefaultAlgorithmChoice.T
+    safetyfallback::Bool
+    DefaultLinearSolver(alg; safetyfallback=true) = new(alg,safetyfallback)
 end
 
 const BLASELTYPES = Union{Float32, Float64, ComplexF32, ComplexF64}
diff --git a/src/default.jl b/src/default.jl
index 545337b3f..acfe8201f 100644
--- a/src/default.jl
+++ b/src/default.jl
@@ -41,6 +41,11 @@ end
     ex = Expr(:if, ex.args...)
 end
 
+# Handle special case of Column-pivoted QR fallback for LU
+function __setfield!(cache::DefaultLinearSolverInit, alg::DefaultLinearSolver, v::LinearAlgebra.QRPivoted)
+    setfield!(cache, :QRFactorizationPivoted, v)
+end
+
 # Legacy fallback
 # For SciML algorithms already using `defaultalg`, all assume square matrix.
 defaultalg(A, b) = defaultalg(A, b, OperatorAssumptions(true))
@@ -352,11 +357,32 @@ end
         kwargs...)
     ex = :()
     for alg in first.(EnumX.symbol_map(DefaultAlgorithmChoice.T))
-        newex = quote
-            sol = SciMLBase.solve!(cache, $(algchoice_to_alg(alg)), args...; kwargs...)
-            SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
-                retcode = sol.retcode,
-                iters = sol.iters, stats = sol.stats)
+        if alg in Symbol.((DefaultAlgorithmChoice.LUFactorization,
+            DefaultAlgorithmChoice.RFLUFactorization,
+            DefaultAlgorithmChoice.MKLLUFactorization,
+            DefaultAlgorithmChoice.AppleAccelerateLUFactorization,
+            DefaultAlgorithmChoice.GenericLUFactorization))
+            newex = quote
+                sol = SciMLBase.solve!(cache, $(algchoice_to_alg(alg)), args...; kwargs...)
+                if sol.retcode === ReturnCode.Failure && alg.safetyfallback
+                    ## TODO: Add verbosity logging here about using the fallback
+                    sol = SciMLBase.solve!(cache, QRFactorization(ColumnNorm()), args...; kwargs...)
+                    SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
+                        retcode = sol.retcode,
+                        iters = sol.iters, stats = sol.stats)
+                else
+                    SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
+                        retcode = sol.retcode,
+                        iters = sol.iters, stats = sol.stats)
+                end
+            end
+        else
+            newex = quote
+                sol = SciMLBase.solve!(cache, $(algchoice_to_alg(alg)), args...; kwargs...)
+                SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
+                    retcode = sol.retcode,
+                    iters = sol.iters, stats = sol.stats)
+            end
         end
         alg_enum = getproperty(LinearSolve.DefaultAlgorithmChoice, alg)
         ex = if ex == :()
diff --git a/test/default_algs.jl b/test/default_algs.jl
index 58f30db50..cf0becfb8 100644
--- a/test/default_algs.jl
+++ b/test/default_algs.jl
@@ -158,4 +158,18 @@ prob = LinearProblem(A, b)
 @test_broken SciMLBase.successful_retcode(solve(prob))
 
 prob2 = LinearProblem(A2, b)
-@test SciMLBase.successful_retcode(solve(prob2))
\ No newline at end of file
+@test SciMLBase.successful_retcode(solve(prob2))
+
+# Column-Pivoted QR fallback on failed LU
+A = [1.0 0 0 0
+     0  1 0 0
+     0  0 1 0
+    0 0 0 0]
+b = rand(4)
+prob = LinearProblem(A, b)
+sol = solve(prob, LinearSolve.DefaultLinearSolver(LinearSolve.DefaultAlgorithmChoice.LUFactorization; safetyfallback=false))
+@test sol.retcode === ReturnCode.Failure
+@test sol.u == zeros(4)
+
+sol = solve(prob)
+@test sol.u ≈ svd(A)\b
diff --git a/test/retcodes.jl b/test/retcodes.jl
index 1e33e8adb..af59de6eb 100644
--- a/test/retcodes.jl
+++ b/test/retcodes.jl
@@ -1,24 +1,13 @@
-using LinearSolve, RecursiveFactorization
+using LinearSolve, LinearAlgebra, RecursiveFactorization
 
 alglist = (
     LUFactorization,
     QRFactorization,
-    DiagonalFactorization,
-    DirectLdiv!,
-    SparspakFactorization,
-    KLUFactorization,
-    UMFPACKFactorization,
     KrylovJL_GMRES,
     GenericLUFactorization,
     RFLUFactorization,
-    LDLtFactorization,
-    BunchKaufmanFactorization,
-    CHOLMODFactorization,
     SVDFactorization,
-    CholeskyFactorization,
     NormalCholeskyFactorization,
-    AppleAccelerateLUFactorization,
-    MKLLUFactorization,
     KrylovJL_CRAIGMR,
     KrylovJL_LSMR
 )
@@ -28,14 +17,23 @@ alglist = (
         A = [2.0 1.0; -1.0 1.0]
         b = [-1.0, 1.0]
         prob = LinearProblem(A, b)
-        linsolve = init(prob, alg)
+        linsolve = init(prob, alg())
         sol = solve!(linsolve)
         @test SciMLBase.successful_retcode(sol.retcode) || sol.retcode == ReturnCode.Default # The latter seems off...
     end
 end
 
+lualgs = (
+    LUFactorization(),
+    QRFactorization(),
+    GenericLUFactorization(),
+    LinearSolve.DefaultLinearSolver(LinearSolve.DefaultAlgorithmChoice.LUFactorization; safetyfallback=false),
+    RFLUFactorization(),
+    NormalCholeskyFactorization(),
+)
 @testset "Failure" begin
-    for alg in alglist
+    for alg in lualgs
+        @show alg
         A = [1.0 1.0; 1.0 1.0]
         b = [-1.0, 1.0]
         prob = LinearProblem(A, b)
@@ -44,3 +42,24 @@ end
         @test !SciMLBase.successful_retcode(sol.retcode)
     end
 end
+
+rankdeficientalgs = (
+    QRFactorization(LinearAlgebra.ColumnNorm()),
+    KrylovJL_GMRES(),
+    SVDFactorization(),
+    KrylovJL_CRAIGMR(),
+    KrylovJL_LSMR(),
+    LinearSolve.DefaultLinearSolver(LinearSolve.DefaultAlgorithmChoice.LUFactorization)
+)
+
+@testset "Rank Deficient Success" begin
+    for alg in rankdeficientalgs
+        @show alg
+        A = [1.0 1.0; 1.0 1.0]
+        b = [-1.0, 1.0]
+        prob = LinearProblem(A, b)
+        linsolve = init(prob, alg)
+        sol = solve!(linsolve)
+        @test SciMLBase.successful_retcode(sol.retcode)
+    end
+end
\ No newline at end of file