diff --git a/libcxx/test/std/input.output/filesystems/class.path/path.member/path.assign/move.pass.cpp b/libcxx/test/std/input.output/filesystems/class.path/path.member/path.assign/move.pass.cpp
--- a/libcxx/test/std/input.output/filesystems/class.path/path.member/path.assign/move.pass.cpp
+++ b/libcxx/test/std/input.output/filesystems/class.path/path.member/path.assign/move.pass.cpp
@@ -8,6 +8,13 @@
 
 // UNSUPPORTED: c++03
 
+// z/OS doesn't currently have runtime linking. This prevents calls to new and
+// delete in the standard library (like the one caused by the allocation in the
+// string constructor) from invoking user-defined versions of those operators
+// (like the ones in count_new.h). Thus, globalMemCounter doesn't see the
+// new we expect it to.
+// UNSUPPORTED: target={{.+}}-zos{{.*}}
+
 // <filesystem>
 
 // class path
diff --git a/libcxx/test/std/input.output/filesystems/class.path/path.member/path.concat.pass.cpp b/libcxx/test/std/input.output/filesystems/class.path/path.member/path.concat.pass.cpp
--- a/libcxx/test/std/input.output/filesystems/class.path/path.member/path.concat.pass.cpp
+++ b/libcxx/test/std/input.output/filesystems/class.path/path.member/path.concat.pass.cpp
@@ -11,6 +11,13 @@
 // These tests require locale for non-char paths
 // UNSUPPORTED: libcpp-has-no-localization
 
+// z/OS doesn't currently have runtime linking. This prevents calls to new and
+// delete in the standard library (like the one during path concatenation
+// caused by basic_string::__grow_by_and_replace) from invoking user-defined
+// versions of those operators (like the ones in count_new.h). Thus,
+// RequireAllocationGuard doesn't see the new we expect it to.
+// UNSUPPORTED: target={{.+}}-zos{{.*}}
+
 // <filesystem>
 
 // class path
diff --git a/libcxx/test/std/input.output/filesystems/class.path/path.member/path.construct/move.pass.cpp b/libcxx/test/std/input.output/filesystems/class.path/path.member/path.construct/move.pass.cpp
--- a/libcxx/test/std/input.output/filesystems/class.path/path.member/path.construct/move.pass.cpp
+++ b/libcxx/test/std/input.output/filesystems/class.path/path.member/path.construct/move.pass.cpp
@@ -8,6 +8,13 @@
 
 // UNSUPPORTED: c++03
 
+// z/OS doesn't currently have runtime linking. This prevents calls to new and
+// delete in the standard library (like the one caused by the allocation in the
+// string constructor) from invoking user-defined versions of those operators
+// (like the ones in count_new.h). Thus, globalMemCounter doesn't see the
+// new we expect it to.
+// UNSUPPORTED: target={{.+}}-zos{{.*}}
+
 // <filesystem>
 
 // class path
diff --git a/libcxx/test/std/localization/locale.categories/category.ctype/facet.ctype.special/facet.ctype.char.dtor/dtor.pass.cpp b/libcxx/test/std/localization/locale.categories/category.ctype/facet.ctype.special/facet.ctype.char.dtor/dtor.pass.cpp
--- a/libcxx/test/std/localization/locale.categories/category.ctype/facet.ctype.special/facet.ctype.char.dtor/dtor.pass.cpp
+++ b/libcxx/test/std/localization/locale.categories/category.ctype/facet.ctype.special/facet.ctype.char.dtor/dtor.pass.cpp
@@ -6,6 +6,12 @@
 //
 //===----------------------------------------------------------------------===//
 
+// z/OS doesn't currently have runtime linking. This prevents calls to new and
+// delete in the standard library (like the one in ctype<char>::~ctype) from
+// invoking user-defined versions of those operators (like the ones in
+// count_new.h). Thus, globalMemCounter doesn't see the delete[] it's expecting.
+// UNSUPPORTED: target={{.+}}-zos{{.*}}
+
 // <locale>
 
 // template <> class ctype<char>
diff --git a/libcxx/test/std/thread/thread.threads/thread.thread.class/thread.thread.constr/F.pass.cpp b/libcxx/test/std/thread/thread.threads/thread.thread.class/thread.thread.constr/F.pass.cpp
--- a/libcxx/test/std/thread/thread.threads/thread.thread.class/thread.thread.constr/F.pass.cpp
+++ b/libcxx/test/std/thread/thread.threads/thread.thread.class/thread.thread.constr/F.pass.cpp
@@ -8,6 +8,12 @@
 //
 // UNSUPPORTED: libcpp-has-no-threads
 
+// z/OS doesn't currently have runtime linking. This prevents calls to new and
+// delete in the standard library (like the one caused by the argument copy in
+// the thread constructor) from invoking user-defined versions of those
+// operators.
+// UNSUPPORTED: target={{.+}}-zos{{.*}}
+
 // <thread>
 
 // class thread