There seems to be two contradicting conventions on web applications. One is that a Dropdown-control also is an action button, meaning that when a user selects an item in the Dropdown control, an event fires and you get to the next step imediately.
The other one is the same as the above except for the action which is moved to a simple button next to the Dropdown control. The workflow here is select, then action. A two step activity instead of just a single step.
Which one is the preferred one?
Neither options seem ideal since the user cannot provide login information right aways and I think that's what users expect when presented with login. If several login options are supported it could default to the most frequently used option and also remember what the user picked previous time allowing users to log in with a single click.
If you have to stick with the two option above I have a feeling that the option without the Login button may do worse initially since users will be scanning the page for a control saying "log in" that is not there. If the page is busy it may take more time to find. If it is a frequently used site and users have time to learn, than the dropdown submitting an action could be a better option since it is less work for the users.
I'd say that for the mobile environment, your first scenario would be preferred in my opinion. Limiting unnecessary actions is always best for on the go browsing that is done on mobile devices.
The only downfall, I suppose, would be in the event of a misclick it might be inconvenient that the event has already triggered - depending on what that event is...
I would say that it depends on the complexity of your application.
A lot of it has to do with how much is loaded and how long it takes. If a selection from a dropdown loads something that takes more than a second or two, I would opt to have the user click the button. This is because if they accidentally select the wrong item in the dropdown, they won't have to wait for it to load data before clicking the right one.
