Enter in Google following bluecard termin auslaenderbehoerde muenchen.Actually, you do not need to enter the exact word-combination which I used. Mind the following keywords: Blue card - appointment - the city name (in our case Munich).
Here make sure that you click on a government website and not on a private company. Let's follow the second result of my search: How do I make an appointment? (Wie vereinbare ich einen Termin?) from www.muenchen.de > City Hall > Make an appointment. Notice that next to each German search result, there is a link "Translate this Page".
We are now on an overview page.
Tip: instead of looking for English mirror, use the German page traanslated into English with the Chrome right click. I noticed that English mirrors sometimes are less developed and not always updated on time. An example of Chrome translation see on the screenshots 4 and 5).
Scroll down to the Appointment: Make an appointment online. Click on it.
Select number of persons. I choose "4" because I want to register the whole family - dad, mum and two kids (screenshot 5).
Choose a free date on the online calendar.
However, it is quite possible that all dates are fully booked for next weeks.
First book any available date, even if this date is behind the validy period of your residence title.
I recommend to book any available date based on your Mitwirkungspflicht - so called "obligation to cooperate". It is not your fault that the Immigration Authority cannot offer free appointments at the moment. By booking any available appointment you have done your best and accepted the earliest date possible. As long as you weren't too late to schedule the application for a bluecard yourself.
Now you have two options:
Option #1: You check online regularly (i.e. daily) in the hope of finding a free date and book it immediately.
The "good news" is that you have to go online very early in the morning: at 7 AM.
Option #2: Hire Migwork. We book an appoinment for you. You will be notified immediately after we have booked the appointment for you (and for your family).
Note: please remember to delete the first booked appointment after you get a suitable date.