diff --git a/stdlib/SparseArrays/src/sparsematrix.jl b/stdlib/SparseArrays/src/sparsematrix.jl
index ea8a8f0b442e2..da08e8a72981c 100644
--- a/stdlib/SparseArrays/src/sparsematrix.jl
+++ b/stdlib/SparseArrays/src/sparsematrix.jl
@@ -563,6 +563,8 @@ SparseMatrixCSC(M::Matrix) = sparse(M)
 SparseMatrixCSC(T::Tridiagonal{Tv}) where Tv = SparseMatrixCSC{Tv,Int}(T)
 function SparseMatrixCSC{Tv,Ti}(T::Tridiagonal) where {Tv,Ti}
     m = length(T.d)
+    m == 0 && return SparseMatrixCSC{Tv,Ti}(0, 0, ones(Ti, 1), Ti[], Tv[])
+    m == 1 && return SparseMatrixCSC{Tv,Ti}(1, 1, Ti[1, 2], Ti[1], Tv[T.d[1]])
 
     colptr = Vector{Ti}(undef, m+1)
     colptr[1] = 1
@@ -593,6 +595,8 @@ end
 SparseMatrixCSC(T::SymTridiagonal{Tv}) where Tv = SparseMatrixCSC{Tv,Int}(T)
 function SparseMatrixCSC{Tv,Ti}(T::SymTridiagonal) where {Tv,Ti}
     m = length(T.dv)
+    m == 0 && return SparseMatrixCSC{Tv,Ti}(0, 0, ones(Ti, 1), Ti[], Tv[])
+    m == 1 && return SparseMatrixCSC{Tv,Ti}(1, 1, Ti[1, 2], Ti[1], Tv[T.dv[1]])
 
     colptr = Vector{Ti}(undef, m+1)
     colptr[1] = 1
@@ -623,7 +627,7 @@ end
 SparseMatrixCSC(B::Bidiagonal{Tv}) where Tv = SparseMatrixCSC{Tv,Int}(B)
 function SparseMatrixCSC{Tv,Ti}(B::Bidiagonal) where {Tv,Ti}
     m = length(B.dv)
-    m == 0 && return SparseMatrixCSC{Tv,Ti}(zeros(Tv, 0, 0))
+    m == 0 && return SparseMatrixCSC{Tv,Ti}(0, 0, ones(Ti, 1), Ti[], Tv[])
 
     colptr = Vector{Ti}(undef, m+1)
     colptr[1] = 1
@@ -652,7 +656,7 @@ end
 SparseMatrixCSC(D::Diagonal{Tv}) where Tv = SparseMatrixCSC{Tv,Int}(D)
 function SparseMatrixCSC{Tv,Ti}(D::Diagonal) where {Tv,Ti}
     m = length(D.diag)
-    return SparseMatrixCSC(m, m, Vector(1:(m+1)), Vector(1:m), Vector{Tv}(D.diag))
+    return SparseMatrixCSC(m, m, Vector(Ti(1):Ti(m+1)), Vector(Ti(1):Ti(m)), Vector{Tv}(D.diag))
 end
 SparseMatrixCSC(M::AbstractMatrix{Tv}) where {Tv} = SparseMatrixCSC{Tv,Int}(M)
 SparseMatrixCSC{Tv}(M::AbstractMatrix{Tv}) where {Tv} = SparseMatrixCSC{Tv,Int}(M)
diff --git a/stdlib/SparseArrays/test/sparse.jl b/stdlib/SparseArrays/test/sparse.jl
index 128bb4735f063..ff955e967b433 100644
--- a/stdlib/SparseArrays/test/sparse.jl
+++ b/stdlib/SparseArrays/test/sparse.jl
@@ -1697,6 +1697,33 @@ end
     S2 = SparseMatrixCSC(D)
     @test Array(D) == Array(S) == Array(S2)
     @test S == S2
+
+    # An issue discovered in #42574 where
+    # SparseMatrixCSC{Tv, Ti}(::Diagonal) ignored Ti
+    D = Diagonal(rand(3))
+    S = SparseMatrixCSC{Float64, Int8}(D)
+    @test S isa SparseMatrixCSC{Float64, Int8}
+end
+
+@testset "Sparse construction with empty/1x1 structured matrices" begin
+    empty = spzeros(0, 0)
+
+    @test sparse(Diagonal(zeros(0, 0))) == empty
+    @test sparse(Bidiagonal(zeros(0, 0), :U)) == empty
+    @test sparse(Bidiagonal(zeros(0, 0), :L)) == empty
+    @test sparse(SymTridiagonal(zeros(0, 0))) == empty
+    @test sparse(Tridiagonal(zeros(0, 0))) == empty
+
+    one_by_one = rand(1,1)
+    sp_one_by_one = sparse(one_by_one)
+
+    @test sparse(Diagonal(one_by_one)) == sp_one_by_one
+    @test sparse(Bidiagonal(one_by_one, :U)) == sp_one_by_one
+    @test sparse(Bidiagonal(one_by_one, :L)) == sp_one_by_one
+    @test sparse(Tridiagonal(one_by_one)) == sp_one_by_one
+
+    s = SymTridiagonal(rand(1), rand(0))
+    @test sparse(s) == s
 end
 
 @testset "error conditions for reshape, and dropdims" begin