diff --git a/commit-graph.c b/commit-graph.c
index fd395e699d..7943da3848 100644
--- a/commit-graph.c
+++ b/commit-graph.c
@@ -2209,7 +2209,8 @@ static void mark_commit_graphs(struct write_commit_graph_context *ctx)
 		struct stat st;
 		struct utimbuf updated_time;
 
-		stat(ctx->commit_graph_filenames_before[i], &st);
+		if (stat(ctx->commit_graph_filenames_before[i], &st) < 0)
+			continue;
 
 		updated_time.actime = st.st_atime;
 		updated_time.modtime = now;
@@ -2250,7 +2251,8 @@ static void expire_commit_graphs(struct write_commit_graph_context *ctx)
 		strbuf_setlen(&path, dirnamelen);
 		strbuf_addstr(&path, de->d_name);
 
-		stat(path.buf, &st);
+		if (stat(path.buf, &st) < 0)
+			continue;
 
 		if (st.st_mtime > expire_time)
 			continue;
diff --git a/object-file.c b/object-file.c
index a81d84132d..d35c1820e7 100644
--- a/object-file.c
+++ b/object-file.c
@@ -2623,8 +2623,12 @@ int read_loose_object(const char *path,
 		goto out;
 	}
 
-	if (unpack_loose_header(&stream, map, mapsize, hdr, sizeof(hdr),
-				NULL) < 0) {
+	switch (unpack_loose_header(&stream, map, mapsize, hdr, sizeof(hdr),
+				    NULL)) {
+	case ULHR_OK:
+		break;
+	case ULHR_BAD:
+	case ULHR_TOO_LONG:
 		error(_("unable to unpack header of %s"), path);
 		goto out;
 	}
diff --git a/t/t0060-path-utils.sh b/t/t0060-path-utils.sh
index 2fe6ae6a4e..aa35350b6f 100755
--- a/t/t0060-path-utils.sh
+++ b/t/t0060-path-utils.sh
@@ -542,7 +542,7 @@ test_lazy_prereq CAN_EXEC_IN_PWD '
 	./git rev-parse
 '
 
-test_expect_success RUNTIME_PREFIX,CAN_EXEC_IN_PWD 'RUNTIME_PREFIX works' '
+test_expect_success !VALGRIND,RUNTIME_PREFIX,CAN_EXEC_IN_PWD 'RUNTIME_PREFIX works' '
 	mkdir -p pretend/bin pretend/libexec/git-core &&
 	echo "echo HERE" | write_script pretend/libexec/git-core/git-here &&
 	cp "$GIT_EXEC_PATH"/git$X pretend/bin/ &&
@@ -550,7 +550,7 @@ test_expect_success RUNTIME_PREFIX,CAN_EXEC_IN_PWD 'RUNTIME_PREFIX works' '
 	echo HERE >expect &&
 	test_cmp expect actual'
 
-test_expect_success RUNTIME_PREFIX,CAN_EXEC_IN_PWD '%(prefix)/ works' '
+test_expect_success !VALGRIND,RUNTIME_PREFIX,CAN_EXEC_IN_PWD '%(prefix)/ works' '
 	mkdir -p pretend/bin &&
 	cp "$GIT_EXEC_PATH"/git$X pretend/bin/ &&
 	git config yes.path "%(prefix)/yes" &&
diff --git a/t/t1006-cat-file.sh b/t/t1006-cat-file.sh
index 1b85207694..dadf3b1458 100755
--- a/t/t1006-cat-file.sh
+++ b/t/t1006-cat-file.sh
@@ -681,7 +681,7 @@ test_expect_success 'cat-file -t and -s on corrupt loose object' '
 
 		# Setup and create the empty blob and its path
 		empty_path=$(git rev-parse --git-path objects/$(test_oid_to_path "$EMPTY_BLOB")) &&
-		git hash-object -w --stdin </dev/null &&
+		empty_blob=$(git hash-object -w --stdin </dev/null) &&
 
 		# Create another blob and its path
 		echo other >other.blob &&
@@ -722,7 +722,13 @@ test_expect_success 'cat-file -t and -s on corrupt loose object' '
 		# content out as-is. Try to make it zlib-invalid.
 		mv -f other.blob "$empty_path" &&
 		test_must_fail git fsck 2>err.fsck &&
-		grep "^error: inflate: data stream error (" err.fsck
+		cat >expect <<-EOF &&
+		error: inflate: data stream error (incorrect header check)
+		error: unable to unpack header of ./$empty_path
+		error: $empty_blob: object corrupt or missing: ./$empty_path
+		EOF
+		grep "^error: " err.fsck >actual &&
+		test_cmp expect actual
 	)
 '
 
diff --git a/t/t1450-fsck.sh b/t/t1450-fsck.sh
index de50c0ea01..ab7f31f1dc 100755
--- a/t/t1450-fsck.sh
+++ b/t/t1450-fsck.sh
@@ -774,10 +774,19 @@ test_expect_success 'fsck finds problems in duplicate loose objects' '
 		# no "-d" here, so we end up with duplicates
 		git repack &&
 		# now corrupt the loose copy
-		file=$(sha1_file "$(git rev-parse HEAD)") &&
+		oid="$(git rev-parse HEAD)" &&
+		file=$(sha1_file "$oid") &&
 		rm "$file" &&
 		echo broken >"$file" &&
-		test_must_fail git fsck
+		test_must_fail git fsck 2>err &&
+
+		cat >expect <<-EOF &&
+		error: inflate: data stream error (incorrect header check)
+		error: unable to unpack header of $file
+		error: $oid: object corrupt or missing: $file
+		EOF
+		grep "^error: " err >actual &&
+		test_cmp expect actual
 	)
 '
 
diff --git a/t/t4202-log.sh b/t/t4202-log.sh
index be07407f85..6e66352558 100755
--- a/t/t4202-log.sh
+++ b/t/t4202-log.sh
@@ -1992,10 +1992,13 @@ test_expect_success GPG 'log --show-signature for merged tag with GPG failure' '
 	git tag -s -m signed_tag_msg signed_tag_fail &&
 	git checkout plain-fail &&
 	git merge --no-ff -m msg signed_tag_fail &&
-	TMPDIR="$(pwd)/bogus" git log --show-signature -n1 plain-fail >actual &&
-	grep "^merged tag" actual &&
-	grep "^No signature" actual &&
-	! grep "^gpg: Signature made" actual
+	if ! test_have_prereq VALGRIND
+	then
+		TMPDIR="$(pwd)/bogus" git log --show-signature -n1 plain-fail >actual &&
+		grep "^merged tag" actual &&
+		grep "^No signature" actual &&
+		! grep "^gpg: Signature made" actual
+	fi
 '
 
 test_expect_success GPGSM 'log --graph --show-signature for merged tag x509' '
diff --git a/t/test-lib.sh b/t/test-lib.sh
index f09e8f3efc..8ba5ca1534 100644
--- a/t/test-lib.sh
+++ b/t/test-lib.sh
@@ -1667,6 +1667,7 @@ test -n "$USE_LIBPCRE2" && test_set_prereq PCRE
 test -n "$USE_LIBPCRE2" && test_set_prereq LIBPCRE2
 test -z "$NO_GETTEXT" && test_set_prereq GETTEXT
 test -n "$SANITIZE_LEAK" && test_set_prereq SANITIZE_LEAK
+test -n "$GIT_VALGRIND_ENABLED" && test_set_prereq VALGRIND
 
 if test -z "$GIT_TEST_CHECK_CACHE_TREE"
 then