diff --git a/reframe/core/pipeline.py b/reframe/core/pipeline.py
index 5cd02efe0a..ef55e5aa4a 100644
--- a/reframe/core/pipeline.py
+++ b/reframe/core/pipeline.py
@@ -46,10 +46,13 @@
 # Dependency kinds
 
 #: Constant to be passed as the ``how`` argument of the
-#: :func:`RegressionTest.depends_on` method. It denotes that test case
+#: :func:`~RegressionTest.depends_on` method. It denotes that test case
 #: dependencies will be explicitly specified by the user.
 #:
 #:  This constant is directly available under the :mod:`reframe` module.
+#:
+#: .. deprecated:: 3.3
+#:    Please use a callable as the ``how`` argument.
 DEPEND_EXACT = 1
 
 #: Constant to be passed as the ``how`` argument of the
@@ -58,6 +61,9 @@
 #: target test that use the same programming environment.
 #:
 #:  This constant is directly available under the :mod:`reframe` module.
+#:
+#: .. deprecated:: 3.3
+#:    Please use a callable as the ``how`` argument.
 DEPEND_BY_ENV = 2
 
 #: Constant to be passed as the ``how`` argument of the
@@ -65,6 +71,9 @@
 #: this test depends on all the test cases of the target test.
 #:
 #:  This constant is directly available under the :mod:`reframe` module.
+#:
+#: .. deprecated:: 3.3
+#:    Please use a callable as the ``how`` argument.
 DEPEND_FULLY = 3
 
 
diff --git a/reframe/utility/__init__.py b/reframe/utility/__init__.py
index df8eb20d9c..46a96c77b6 100644
--- a/reframe/utility/__init__.py
+++ b/reframe/utility/__init__.py
@@ -565,10 +565,12 @@ def find_modules(substr, environ_mapping=None):
 
     .. code:: python
 
-       @rfm.parameterized_test(*find_modules('netcdf'))
+       @rfm.simple_test
        class MyTest(rfm.RegressionTest):
-           def __init__(self, s, e, m):
-               self.descr = f'{s}, {e}, {m}'
+           module_info = parameter(find_modules('netcdf'))
+
+           def __init__(self):
+               s, e, m = self.module_info
                self.valid_systems = [s]
                self.valid_prog_environs = [e]
                self.modules = [m]
@@ -583,15 +585,17 @@ def __init__(self, s, e, m):
     .. code:: python
 
        my_find_modules = functools.partial(find_modules, environ_mapping={
-           r'.*CrayGNU.*': {'PrgEnv-gnu'},
-           r'.*CrayIntel.*': {'PrgEnv-intel'},
-           r'.*CrayCCE.*': {'PrgEnv-cray'}
+           r'.*CrayGNU.*': 'PrgEnv-gnu',
+           r'.*CrayIntel.*': 'PrgEnv-intel',
+           r'.*CrayCCE.*': 'PrgEnv-cray'
        })
 
-       @rfm.parameterized_test(*my_find_modules('GROMACS'))
+       @rfm.simple_test
        class MyTest(rfm.RegressionTest):
-           def __init__(self, s, e, m):
-               self.descr = f'{s}, {e}, {m}'
+           module_info = parameter(my_find_modules('GROMACS'))
+
+           def __init__(self):
+               s, e, m = self.module_info
                self.valid_systems = [s]
                self.valid_prog_environs = [e]
                self.modules = [m]