diff --git a/tests/compat.sh b/tests/compat.sh
index d7a91b47e..20f2dbda6 100755
--- a/tests/compat.sh
+++ b/tests/compat.sh
@@ -588,7 +588,18 @@ add_mbedtls_ciphersuites()
 # o_check_ciphersuite STANDARD_CIPHER_SUITE
 o_check_ciphersuite()
 {
-    if [ "${O_SUPPORT_ECDH}" = "NO" ]; then
+    # skip DTLS when lack of support was declared
+    if test "$OSSL_NO_DTLS" -gt 0 && is_dtls "$MODE"; then
+        SKIP_NEXT_="YES"
+    fi
+
+    # skip DTLS 1.2 is support was not detected
+    if [ "$O_SUPPORT_DTLS12" = "NO" -a "$MODE" = "dtls12" ]; then
+        SKIP_NEXT="YES"
+    fi
+
+    # skip static ECDH when OpenSSL doesn't support it
+    if [ "${O_SUPPORT_STATIC_ECDH}" = "NO" ]; then
         case "$1" in
             *ECDH_*) SKIP_NEXT="YES"
         esac
@@ -665,10 +676,25 @@ setup_arguments()
     esac
 
     case $($OPENSSL ciphers ALL) in
-        *ECDH-ECDSA*|*ECDH-RSA*) O_SUPPORT_ECDH="YES";;
-        *) O_SUPPORT_ECDH="NO";;
+        *ECDH-ECDSA*|*ECDH-RSA*) O_SUPPORT_STATIC_ECDH="YES";;
+        *) O_SUPPORT_STATIC_ECDH="NO";;
     esac
 
+    case $($OPENSSL ciphers ALL) in
+        *DES-CBC-*) O_SUPPORT_SINGLE_DES="YES";;
+        *) O_SUPPORT_SINGLE_DES="NO";;
+    esac
+
+    # OpenSSL <1.0.2 doesn't support DTLS 1.2. Check if OpenSSL
+    # supports -dtls1_2 from the s_server help. (The s_client
+    # help isn't accurate as of 1.0.2g: it supports DTLS 1.2
+    # but doesn't list it. But the s_server help seems to be
+    # accurate.)
+    O_SUPPORT_DTLS12="NO"
+    if $OPENSSL s_server -help 2>&1 | grep -q "^ *-dtls1_2 "; then
+        O_SUPPORT_DTLS12="YES"
+    fi
+
     if [ "X$VERIFY" = "XYES" ];
     then
         M_SERVER_ARGS="$M_SERVER_ARGS ca_file=data_files/test-ca_cat12.crt auth_mode=required"
@@ -1109,19 +1135,6 @@ for MODE in $MODES; do
 
                 [Oo]pen*)
 
-                    if test "$OSSL_NO_DTLS" -gt 0 && is_dtls "$MODE"; then
-                        continue;
-                    fi
-
-                    # OpenSSL <1.0.2 doesn't support DTLS 1.2. Check if OpenSSL
-                    # supports $O_MODE from the s_server help. (The s_client
-                    # help isn't accurate as of 1.0.2g: it supports DTLS 1.2
-                    # but doesn't list it. But the s_server help seems to be
-                    # accurate.)
-                    if ! $OPENSSL s_server -help 2>&1 | grep -q "^ *-$O_MODE "; then
-                        continue;
-                    fi
-
                     reset_ciphersuites
                     add_common_ciphersuites
                     add_openssl_ciphersuites
diff --git a/tests/scripts/all.sh b/tests/scripts/all.sh
index 985d14d39..28009d56a 100755
--- a/tests/scripts/all.sh
+++ b/tests/scripts/all.sh
@@ -1220,14 +1220,19 @@ component_test_full_cmake_gcc_asan () {
     msg "test: main suites (inc. selftests) (full config, ASan build)"
     make test
 
-    msg "test: selftest (ASan build)" # ~ 10s
+    msg "test: selftest (full config, ASan build)" # ~ 10s
     programs/test/selftest
 
     msg "test: ssl-opt.sh (full config, ASan build)"
     tests/ssl-opt.sh
 
-    msg "test: compat.sh (full config, ASan build)"
-    tests/compat.sh
+    # Note: the next two invocations cover all compat.sh test cases.
+    # We should use the same here and in basic-build-test.sh.
+    msg "test: compat.sh: default version (full config, ASan build)"
+    tests/compat.sh -e 'ARIA\|CHACHA'
+
+    msg "test: compat.sh: next: ARIA, Chacha (full config, ASan build)"
+    env OPENSSL="$OPENSSL_NEXT" tests/compat.sh -e '^$' -f 'ARIA\|CHACHA'
 
     msg "test: context-info.sh (full config, ASan build)" # ~ 15 sec
     tests/context-info.sh
@@ -1241,19 +1246,24 @@ component_test_full_cmake_gcc_asan_new_bignum () {
     CC=gcc cmake -D CMAKE_BUILD_TYPE:String=Asan .
     make
 
-    msg "test: main suites (inc. selftests) (full config, ASan build)"
+    msg "test: main suites (inc. selftests) (full config, new bignum, ASan)"
     make test
 
-    msg "test: selftest (ASan build)" # ~ 10s
+    msg "test: selftest (full config, new bignum, ASan)" # ~ 10s
     programs/test/selftest
 
-    msg "test: ssl-opt.sh (full config, ASan build)"
+    msg "test: ssl-opt.sh (full config, new bignum, ASan)"
     tests/ssl-opt.sh
 
-    msg "test: compat.sh (full config, ASan build)"
-    tests/compat.sh
+    # Note: the next two invocations cover all compat.sh test cases.
+    # We should use the same here and in basic-build-test.sh.
+    msg "test: compat.sh: default version (full config, new bignum, ASan)"
+    tests/compat.sh -e 'ARIA\|CHACHA'
 
-    msg "test: context-info.sh (full config, ASan build)" # ~ 15 sec
+    msg "test: compat.sh: next: ARIA, Chacha (full config, new bignum, ASan)"
+    env OPENSSL="$OPENSSL_NEXT" tests/compat.sh -e '^$' -f 'ARIA\|CHACHA'
+
+    msg "test: context-info.sh (full config, new bignum, ASan)" # ~ 15 sec
     tests/context-info.sh
 }
 
@@ -2171,12 +2181,6 @@ component_test_full_cmake_clang () {
 
     msg "test: ssl-opt.sh default, ECJPAKE, SSL async (full config)" # ~ 1s
     tests/ssl-opt.sh -f 'Default\|ECJPAKE\|SSL async private'
-
-    msg "test: compat.sh NULL (full config)" # ~ 2 min
-    tests/compat.sh -e '^$' -f 'NULL'
-
-    msg "test: compat.sh ARIA + ChachaPoly"
-    env OPENSSL="$OPENSSL_NEXT" tests/compat.sh -e '^$' -f 'ARIA\|CHACHA'
 }
 
 skip_suites_without_constant_flow () {
@@ -2620,13 +2624,12 @@ component_test_no_psa_crypto_full_cmake_asan() {
     msg "test: ssl-opt.sh (full minus PSA crypto)"
     tests/ssl-opt.sh
 
-    msg "test: compat.sh default (full minus PSA crypto)"
-    tests/compat.sh
+    # Note: the next two invocations cover all compat.sh test cases.
+    # We should use the same here and in basic-build-test.sh.
+    msg "test: compat.sh: default version (full minus PSA crypto)"
+    tests/compat.sh -e 'ARIA\|CHACHA'
 
-    msg "test: compat.sh NULL (full minus PSA crypto)"
-    tests/compat.sh -f 'NULL'
-
-    msg "test: compat.sh ARIA + ChachaPoly (full minus PSA crypto)"
+    msg "test: compat.sh: next: ARIA, Chacha (full minus PSA crypto)"
     env OPENSSL="$OPENSSL_NEXT" tests/compat.sh -e '^$' -f 'ARIA\|CHACHA'
 }
 
diff --git a/tests/scripts/basic-build-test.sh b/tests/scripts/basic-build-test.sh
index 52617541d..d2e955f1e 100755
--- a/tests/scripts/basic-build-test.sh
+++ b/tests/scripts/basic-build-test.sh
@@ -103,11 +103,7 @@ echo
 echo '################ compat.sh ################'
 {
     echo '#### compat.sh: Default versions'
-    sh compat.sh
-    echo
-
-    echo '#### compat.sh: null cipher'
-    sh compat.sh -e '^$' -f 'NULL'
+    sh compat.sh -e 'ARIA\|CHACHA'
     echo
 
     echo '#### compat.sh: next (ARIA, ChaCha)'