Q:

How to refer to weekdays

I am curious what is the most correct way to communicate what you do during the week.  I have seen in a dictionary that it is possible to use:

Am Montag ... 

And there is the variant:

Montags ...

Are there any distinctions between these two? 

A:

If you use for example: 

Montags höre ich Musik. 

This implies that you normally hear music every monday.    
Otherwise if you use for example: 

Am Montag höre ich Musik. 

You hear music this monday.  
The distinction between these two options is that 'Wochentag + s' implies that you normally always do an specific action on this day.  

A:

The easier part is montags.  We use it for activities that we do or that happen regularly or at least frequently on Mondays. 

Montags bin ich immer so müde. 
  Donnerstags gehe ich zum Gesangsunterricht. 
  Sonntags bleibe ich gerne bis Mittag im Bett. 
  Bis 2015 erschien Der Spiegel immer montags.   

Am Montag is more difficult.  You need to take care of the context.  Often it means a specific day:

Wollen wir nächste Woche ins Kino gehen? 
  Keine Zeit - am Montag ist Elternabend in der Schule, am Dienstag muss ich meine Steuerklärung machen, am Mittwoch packen wir die Koffer, und am Donnerstag fahren wir übers Wochenende zu meinen Schwiegereltern. 

But it can also be used as a synonym to montags.  Look e.g. at this message from Der Spiegel.  In the text body they write

Der SPIEGEL erscheint nun immer samstags. 

But the breadcrumb navigation in the header reads

Der SPIEGEL erscheint jetzt am Samstag. 

Or if someone talks about her childhood you might hear her saying

Am Sonntag gingen wir immer in die Kirche zum Gottesdienst. 

and it means that they went to church (almost) every Sunday.