diff --git a/tests/tcg/s390x/Makefile.softmmu-target b/tests/tcg/s390x/Makefile.softmmu-target
index 725b6c598db..6d8bf299b28 100644
--- a/tests/tcg/s390x/Makefile.softmmu-target
+++ b/tests/tcg/s390x/Makefile.softmmu-target
@@ -1,11 +1,20 @@
 S390X_SRC=$(SRC_PATH)/tests/tcg/s390x
 VPATH+=$(S390X_SRC)
 QEMU_OPTS=-action panic=exit-failure -kernel
+LINK_SCRIPT=$(S390X_SRC)/softmmu.ld
+LDFLAGS=-nostdlib -static -Wl,-T$(LINK_SCRIPT)
 
-%: %.S
-	$(CC) -march=z13 -m64 -nostdlib -static -Wl,-Ttext=0 \
-		-Wl,--build-id=none $< -o $@
+%.o: %.S
+	$(CC) -march=z13 -m64 -c $< -o $@
+
+%: %.o $(LINK_SCRIPT)
+	$(CC) $< -o $@ $(LDFLAGS)
 
 TESTS += unaligned-lowcore
 TESTS += bal
 TESTS += sam
+
+include $(S390X_SRC)/pgm-specification.mak
+$(PGM_SPECIFICATION_TESTS): pgm-specification-softmmu.o
+$(PGM_SPECIFICATION_TESTS): LDFLAGS+=pgm-specification-softmmu.o
+TESTS += $(PGM_SPECIFICATION_TESTS)
diff --git a/tests/tcg/s390x/Makefile.target b/tests/tcg/s390x/Makefile.target
index cf93b966862..1002ab79886 100644
--- a/tests/tcg/s390x/Makefile.target
+++ b/tests/tcg/s390x/Makefile.target
@@ -2,6 +2,9 @@ S390X_SRC=$(SRC_PATH)/tests/tcg/s390x
 VPATH+=$(S390X_SRC)
 CFLAGS+=-march=zEC12 -m64
 
+%.o: %.c
+	$(CC) $(CFLAGS) $(EXTRA_CFLAGS) -c $< -o $@
+
 config-cc.mak: Makefile
 	$(quiet-@)( \
 	    $(call cc-option,-march=z14, CROSS_CC_HAS_Z14); \
@@ -33,6 +36,11 @@ TESTS+=chrl
 cdsg: CFLAGS+=-pthread
 cdsg: LDFLAGS+=-pthread
 
+include $(S390X_SRC)/pgm-specification.mak
+$(PGM_SPECIFICATION_TESTS): pgm-specification-user.o
+$(PGM_SPECIFICATION_TESTS): LDFLAGS+=pgm-specification-user.o
+TESTS += $(PGM_SPECIFICATION_TESTS)
+
 Z13_TESTS=vistr
 $(Z13_TESTS): CFLAGS+=-march=z13 -O2
 TESTS+=$(Z13_TESTS)
diff --git a/tests/tcg/s390x/br-odd.S b/tests/tcg/s390x/br-odd.S
new file mode 100644
index 00000000000..2fae47a9e34
--- /dev/null
+++ b/tests/tcg/s390x/br-odd.S
@@ -0,0 +1,16 @@
+/*
+ * Test BRanching to a non-mapped odd address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    lgrl %r1,odd_addr
+    br %r1
+
+    .align 8
+odd_addr:
+    .quad 0xDDDDDDDDDDDDDDDD
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,0xDDDDDDDDDDDDDDDD
diff --git a/tests/tcg/s390x/cgrl-unaligned.S b/tests/tcg/s390x/cgrl-unaligned.S
new file mode 100644
index 00000000000..164d68f2e64
--- /dev/null
+++ b/tests/tcg/s390x/cgrl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test CGRL with a non-doubleword aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    cgrl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .long 0
+unaligned:
+    .quad 0
diff --git a/tests/tcg/s390x/clrl-unaligned.S b/tests/tcg/s390x/clrl-unaligned.S
new file mode 100644
index 00000000000..182b1b6462f
--- /dev/null
+++ b/tests/tcg/s390x/clrl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test CLRL with a non-word aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    clrl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .short 0
+unaligned:
+    .long 0
diff --git a/tests/tcg/s390x/crl-unaligned.S b/tests/tcg/s390x/crl-unaligned.S
new file mode 100644
index 00000000000..b86fbe0ef3f
--- /dev/null
+++ b/tests/tcg/s390x/crl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test CRL with a non-word aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    crl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .short 0
+unaligned:
+    .long 0
diff --git a/tests/tcg/s390x/ex-odd.S b/tests/tcg/s390x/ex-odd.S
new file mode 100644
index 00000000000..4e42a47df34
--- /dev/null
+++ b/tests/tcg/s390x/ex-odd.S
@@ -0,0 +1,17 @@
+/*
+ * Test EXECUTing a non-mapped odd address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    lgrl %r1,odd_addr
+fail:
+    ex 0,0(%r1)
+
+    .align 8
+odd_addr:
+    .quad 0xDDDDDDDDDDDDDDDD
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,fail
diff --git a/tests/tcg/s390x/lgrl-unaligned.S b/tests/tcg/s390x/lgrl-unaligned.S
new file mode 100644
index 00000000000..ef8d51d47c9
--- /dev/null
+++ b/tests/tcg/s390x/lgrl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test LGRL from a non-doubleword aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    lgrl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .long 0
+unaligned:
+    .quad 0
diff --git a/tests/tcg/s390x/llgfrl-unaligned.S b/tests/tcg/s390x/llgfrl-unaligned.S
new file mode 100644
index 00000000000..c9b4eeaecf5
--- /dev/null
+++ b/tests/tcg/s390x/llgfrl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test LLGFRL from a non-word aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    llgfrl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .short 0
+unaligned:
+    .long 0
diff --git a/tests/tcg/s390x/lpswe-unaligned.S b/tests/tcg/s390x/lpswe-unaligned.S
new file mode 100644
index 00000000000..989f249a6ad
--- /dev/null
+++ b/tests/tcg/s390x/lpswe-unaligned.S
@@ -0,0 +1,18 @@
+/*
+ * Test LPSWE from a non-doubleword aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    larl %r1,unaligned
+fail:
+    lpswe 0(%r1)
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,fail
+    .long 0
+unaligned:
+    .quad 0
diff --git a/tests/tcg/s390x/lrl-unaligned.S b/tests/tcg/s390x/lrl-unaligned.S
new file mode 100644
index 00000000000..11eb07f93a7
--- /dev/null
+++ b/tests/tcg/s390x/lrl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test LRL from a non-word aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    lrl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .short 0
+unaligned:
+    .long 0
diff --git a/tests/tcg/s390x/pgm-specification-softmmu.S b/tests/tcg/s390x/pgm-specification-softmmu.S
new file mode 100644
index 00000000000..d534f4e505d
--- /dev/null
+++ b/tests/tcg/s390x/pgm-specification-softmmu.S
@@ -0,0 +1,40 @@
+/*
+ * Common softmmu code for specification exception testing.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .section .head
+    .org 0x8d
+ilc:
+    .org 0x8e
+program_interruption_code:
+    .org 0x150
+program_old_psw:
+    .org 0x1D0                         /* program new PSW */
+    .quad 0x180000000,pgm              /* 64-bit mode */
+    .org 0x200                         /* lowcore padding */
+
+    .globl _start
+_start:
+    lpswe test_psw
+
+pgm:
+    chhsi program_interruption_code,0x6          /* PGM_SPECIFICATION? */
+    jne failure
+    lg %r0,expected_old_psw+8                    /* ilc adjustment */
+    llgc %r1,ilc
+    agr %r0,%r1
+    stg %r0,expected_old_psw+8
+    clc expected_old_psw(16),program_old_psw     /* correct location? */
+    jne failure
+    lpswe success_psw
+failure:
+    lpswe failure_psw
+
+    .align 8
+test_psw:
+    .quad 0x180000000,test             /* 64-bit mode */
+success_psw:
+    .quad 0x2000180000000,0xfff        /* see is_special_wait_psw() */
+failure_psw:
+    .quad 0x2000180000000,0            /* disabled wait */
diff --git a/tests/tcg/s390x/pgm-specification-user.c b/tests/tcg/s390x/pgm-specification-user.c
new file mode 100644
index 00000000000..9ee6907b7c2
--- /dev/null
+++ b/tests/tcg/s390x/pgm-specification-user.c
@@ -0,0 +1,37 @@
+/*
+ * Common user code for specification exception testing.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+#include <assert.h>
+#include <signal.h>
+#include <stdlib.h>
+#include <string.h>
+#include <unistd.h>
+
+extern void test(void);
+extern long expected_old_psw[2];
+
+static void handle_sigill(int sig, siginfo_t *info, void *ucontext)
+{
+    if ((long)info->si_addr != expected_old_psw[1]) {
+        _exit(EXIT_FAILURE);
+    }
+    _exit(EXIT_SUCCESS);
+}
+
+int main(void)
+{
+    struct sigaction act;
+    int err;
+
+    memset(&act, 0, sizeof(act));
+    act.sa_sigaction = handle_sigill;
+    act.sa_flags = SA_SIGINFO;
+    err = sigaction(SIGILL, &act, NULL);
+    assert(err == 0);
+
+    test();
+
+    return EXIT_FAILURE;
+}
diff --git a/tests/tcg/s390x/pgm-specification.mak b/tests/tcg/s390x/pgm-specification.mak
new file mode 100644
index 00000000000..2999aee26e6
--- /dev/null
+++ b/tests/tcg/s390x/pgm-specification.mak
@@ -0,0 +1,15 @@
+# SPDX-License-Identifier: GPL-2.0-or-later
+# List of specification exception tests.
+# Shared between the softmmu and the user makefiles.
+PGM_SPECIFICATION_TESTS = \
+	br-odd \
+	cgrl-unaligned \
+	clrl-unaligned \
+	crl-unaligned \
+	ex-odd \
+	lgrl-unaligned \
+	llgfrl-unaligned \
+	lpswe-unaligned \
+	lrl-unaligned \
+	stgrl-unaligned \
+	strl-unaligned
diff --git a/tests/tcg/s390x/softmmu.ld b/tests/tcg/s390x/softmmu.ld
new file mode 100644
index 00000000000..ea944eaa3cb
--- /dev/null
+++ b/tests/tcg/s390x/softmmu.ld
@@ -0,0 +1,20 @@
+/*
+ * Linker script for the softmmu test kernels.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+
+ENTRY(_start)
+
+SECTIONS {
+    . = 0;
+
+    .text : {
+        *(.head)
+        *(.text)
+    }
+
+    /DISCARD/ : {
+        *(*)
+    }
+}
diff --git a/tests/tcg/s390x/stgrl-unaligned.S b/tests/tcg/s390x/stgrl-unaligned.S
new file mode 100644
index 00000000000..32df37780ad
--- /dev/null
+++ b/tests/tcg/s390x/stgrl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test STGRL to a non-doubleword aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    stgrl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .long 0
+unaligned:
+    .quad 0
diff --git a/tests/tcg/s390x/strl-unaligned.S b/tests/tcg/s390x/strl-unaligned.S
new file mode 100644
index 00000000000..1d248819f02
--- /dev/null
+++ b/tests/tcg/s390x/strl-unaligned.S
@@ -0,0 +1,16 @@
+/*
+ * Test STRL to a non-word aligned address.
+ *
+ * SPDX-License-Identifier: GPL-2.0-or-later
+ */
+    .globl test
+test:
+    strl %r1,unaligned
+
+    .align 8
+    .globl expected_old_psw
+expected_old_psw:
+    .quad 0x180000000,test
+    .short 0
+unaligned:
+    .long 0
