diff --git a/ext/ArrayLayoutsSparseArraysExt.jl b/ext/ArrayLayoutsSparseArraysExt.jl
index 21d4a64..3d74958 100644
--- a/ext/ArrayLayoutsSparseArraysExt.jl
+++ b/ext/ArrayLayoutsSparseArraysExt.jl
@@ -1,25 +1,69 @@
 module ArrayLayoutsSparseArraysExt
 
 using ArrayLayouts
-using ArrayLayouts: _copyto!
+using ArrayLayouts: _copyto!, Factorization
 using SparseArrays
 # Specifying the full namespace is necessary because of https://github.com/JuliaLang/julia/issues/48533
 # See https://github.com/JuliaStats/LogExpFunctions.jl/pull/63
 import ArrayLayouts.LinearAlgebra
 
-import Base: copyto!
+import Base: copyto!, \, /
 
 # ambiguity from sparsematrix.jl
 copyto!(dest::LayoutMatrix, src::SparseArrays.AbstractSparseMatrixCSC) =
-	_copyto!(dest, src)
+    _copyto!(dest, src)
 
 copyto!(dest::SubArray{<:Any,2,<:LayoutMatrix}, src::SparseArrays.AbstractSparseMatrixCSC) =
-	_copyto!(dest, src)
+    _copyto!(dest, src)
 
 @inline LinearAlgebra.dot(a::LayoutArray{<:Number}, b::SparseArrays.SparseVectorUnion{<:Number}) =
-	ArrayLayouts.dot(a,b)
+    ArrayLayouts.dot(a,b)
 
 @inline LinearAlgebra.dot(a::SparseArrays.SparseVectorUnion{<:Number}, b::LayoutArray{<:Number}) =
-	ArrayLayouts.dot(a,b)
+    ArrayLayouts.dot(a,b)
+
+# disambiguiate sparse matrix dispatches
+macro _layoutldivsp(Typ)
+    ret = quote
+        (\)(x::SparseArrays.AbstractSparseMatrixCSC, A::$Typ; kwds...) = ArrayLayouts.ldiv(x,A; kwds...)
+        (/)(x::SparseArrays.AbstractSparseMatrixCSC, A::$Typ; kwds...) = ArrayLayouts.ldiv(x,A; kwds...)
+    end
+    esc(ret)
+end
+
+macro layoutldivsp(Typ)
+    esc(quote
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp $Typ
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UpperTriangular{T, <:$Typ{T}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitUpperTriangular{T, <:$Typ{T}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.LowerTriangular{T, <:$Typ{T}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitLowerTriangular{T, <:$Typ{T}} where T
+
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UpperTriangular{T, <:SubArray{T,2,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitUpperTriangular{T, <:SubArray{T,2,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.LowerTriangular{T, <:SubArray{T,2,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitLowerTriangular{T, <:SubArray{T,2,<:$Typ{T}}} where T
+
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UpperTriangular{T, <:LinearAlgebra.Adjoint{T,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitUpperTriangular{T, <:LinearAlgebra.Adjoint{T,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.LowerTriangular{T, <:LinearAlgebra.Adjoint{T,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitLowerTriangular{T, <:LinearAlgebra.Adjoint{T,<:$Typ{T}}} where T
+
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UpperTriangular{T, <:LinearAlgebra.Transpose{T,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitUpperTriangular{T, <:LinearAlgebra.Transpose{T,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.LowerTriangular{T, <:LinearAlgebra.Transpose{T,<:$Typ{T}}} where T
+        ArrayLayoutsSparseArraysExt.@_layoutldivsp LinearAlgebra.UnitLowerTriangular{T, <:LinearAlgebra.Transpose{T,<:$Typ{T}}} where T
+    end)
+end
+
+@_layoutldivsp LayoutVector
+
+macro layoutmatrixsp(Typ)
+    esc(quote
+        ArrayLayoutsSparseArraysExt.@layoutldivsp $Typ
+    end)
+end
+
+@layoutmatrixsp LayoutMatrix
 
 end
diff --git a/src/ldiv.jl b/src/ldiv.jl
index 77ffd94..c9ec3a5 100644
--- a/src/ldiv.jl
+++ b/src/ldiv.jl
@@ -151,6 +151,8 @@ macro _layoutldiv(Typ)
         LinearAlgebra.ldiv!(A::LU, x::$Typ; kwds...) = ArrayLayouts.ldiv!(A,x; kwds...)
         LinearAlgebra.ldiv!(A::Cholesky, x::$Typ; kwds...) = ArrayLayouts.ldiv!(A,x; kwds...)
         LinearAlgebra.ldiv!(A::LinearAlgebra.QRCompactWY, x::$Typ; kwds...) = ArrayLayouts.ldiv!(A,x; kwds...)
+        # Type restriction to disambiguiate calls, see https://github.com/JuliaArrays/BlockArrays.jl/issues/319
+        LinearAlgebra.ldiv!(A::LinearAlgebra.QRCompactWY{T2}, x::$Typ{T2}; kwds...) where T2<:BlasFloat = ArrayLayouts.ldiv!(A,x; kwds...)
 
         LinearAlgebra.ldiv!(A::Bidiagonal, B::$Typ; kwds...) = ArrayLayouts.ldiv!(A,B; kwds...)
 
diff --git a/test/test_layoutarray.jl b/test/test_layoutarray.jl
index ee66e17..4f77376 100644
--- a/test/test_layoutarray.jl
+++ b/test/test_layoutarray.jl
@@ -140,11 +140,7 @@ MemoryLayout(::Type{MyVector}) = DenseColumnMajor()
                 else
                     @test ldiv!(lu(A), MyVector(copy(c))) ≈ A \ c
                 end
-                if VERSION < v"1.9" || VERSION >= v"1.10-"
-                    @test_throws ErrorException ldiv!(qr(A), MyVector(copy(c)))
-                else
-                    @test_throws MethodError ldiv!(qr(A), MyVector(copy(c)))
-                end
+                @test_throws ErrorException ldiv!(qr(A), MyVector(copy(c))) # Missing materialize! overload
                 @test_throws ErrorException ldiv!(eigen(randn(5,5)), c)
                 @test ArrayLayouts.ldiv!(svd(A.A), copy(c)) ≈ ArrayLayouts.ldiv!(similar(c), svd(A.A), c) ≈ A \ c
                 if VERSION ≥ v"1.8"
@@ -213,6 +209,7 @@ MemoryLayout(::Type{MyVector}) = DenseColumnMajor()
 
         @testset "layoutldiv" begin
             A = MyMatrix(randn(5,5))
+            Asym = A'*A
             x = randn(5)
             X = randn(5,5)
             t = view(randn(10),[1,3,4,6,7])
@@ -221,8 +218,11 @@ MemoryLayout(::Type{MyVector}) = DenseColumnMajor()
             T̃ = copy(T)
             B = Bidiagonal(randn(5),randn(4),:U)
             D = Diagonal(randn(5))
+            S = sprand(5, 5, 0.8)
+
             @test ldiv!(A, copy(x)) ≈ A\x
             @test A\t ≈ A\t̃
+            @test A\t ≈ A\SparseVector(t̃)
             # QR is not general enough
             @test_broken ldiv!(A, t) ≈ A\t
             @test ldiv!(A, copy(X)) ≈ A\X
@@ -238,6 +238,32 @@ MemoryLayout(::Type{MyVector}) = DenseColumnMajor()
             @test A\MyVector(x) ≈ A\x
             @test A\MyMatrix(X) ≈ A\X
 
+            # Regression for https://github.com/JuliaArrays/BlockArrays.jl/issues/319
+            @test qr(A)\MyVector(x) ≈ A\x
+            @test ldiv!(qr(A), MyVector(copy(x))) ≈ A\x
+            @test qr(A)\MyMatrix(X) ≈ A\X
+            @test ldiv!(qr(A), MyMatrix(copy(X))) ≈ A\X
+
+            @test lu(A)\MyVector(x) ≈ A\x
+            @test ldiv!(lu(A), MyVector(copy(x))) ≈ A\x
+            @test lu(A)\MyMatrix(X) ≈ A\X
+            @test ldiv!(lu(A), MyMatrix(copy(X))) ≈ A\X
+
+            @test cholesky(Asym)\MyVector(x) ≈ Asym\x
+            @test ldiv!(cholesky(Asym), MyVector(copy(x))) ≈ Asym\x
+            @test cholesky(Asym)\MyMatrix(X) ≈ Asym\X
+            @test ldiv!(cholesky(Asym), MyMatrix(copy(X))) ≈ Asym\X
+
+            @test S\MyVector(x) ≈ S\x
+            @test S\MyMatrix(X) ≈ S\X
+            @test MyMatrix(S)\MyVector(x) ≈ S\x
+            @test MyMatrix(S)\MyMatrix(X) ≈ S\X
+
+            @test MyVector(x)'/MyMatrix(S) ≈ x'/S
+            @test X/MyMatrix(S) ≈ X/S
+            @test MyVector(x)'/MyMatrix(S) ≈ x'/S
+            @test X/MyMatrix(S) ≈ X/S
+
             if VERSION >= v"1.9"
                 @test A/A ≈ A.A / A.A
                 @test x' / A ≈ x' / A.A