Apply only database updates for a single module update.php and drush up --no-core both list several database updates pending across several modules. I'm being asked to only apply the database updates for a few specific modules because it is desired to do just these modules and test things in relation to that before running any of the other updates.

drush seems to only give me the option to apply all of the pending updates as does the update.php script. Is there a way to apply only the pending database updates for a particular module when there are pending updates for multiple?
Drush allows you to just update a module you specify:

drush up module_machine_name


If you just want to apply pending database updates, for the modules that have already been updated:

drush updatedb


This case is relevant when updating the code base through git, or for multisite installations. 
In your specific case, this might be what you were looking for - just apply database updates (using this command: for all modules that require database updates), but don't update any additional codebase. 

If you want to just apply the updates for a certain module, you can do this also manually. The updates can be found in the module.install file:

function modulename_update_prioritynumber
(e.g. devel_update_7000)


The priority number determines the order of the updates to be applied. 
You can manually execute the code for the updates using drush:

drush eval "the update code"


for example:

drush eval "variable_del('dev_query');
drush @yoursite eval "variable_del('dev_query');


(@yoursite added if bootstrapping is needed for multisite installations)

This example will delete the variable dev_query from the variable database table (from the function devel_update_7002). 

For more discussion on this, see: drush updatedb for a single module - especially the answer from moshe weitzman, the drush maintainer, about applying individual updates. I'd recommend applying the updates of the module you are concerned about over uncommenting the updatedb statements in other modules - as the other answer suggests.