This basic expression of gratitude can be written in three different ways: 1. Upper case as Dankeschön, one word; 2. Two words upper case Danke and lower case schön; or 3. as two words danke schön in lower case. But which of these are correct?
The uppercase single word Dankeschön is a neuter noun, and should actually only be written thus when clearly used as a noun in a sentence:
Das Lied ist ein Dankeschön an Menschen, die die Sporties inspiriert haben.
The song is a thank-you to people who have inspired the "Sporties."
Captions 26-27, Sportfreunde Stiller: Neues Album
Und ganz herzliches Dankeschön auf jeden Fall.
And very heartfelt thanks in any case.
Caption 20, Tierfreund Mario Barth: Der Tätowierer der Stars
The more common greeting of Danke schön / danke schön is written as two words, and in most cases is written lower case (except when starting a sentence, of course):
Das bestelle ich später, danke schön.
I'll order that later, thank you.
Caption 17, Abendessen: mit Marko
The upper case exception Danke schön, which is recommended (but not required) by Duden, can be used when expressions of gratitude are referred to with the verb sagen in a sentence, in which case the expression is handled grammatically as a noun phrase:
Nächste Woche geht es wieder weiter und ich sage Danke schön und Auf Wiedersehen.
We'll continue next week, and I'd like to say thank you and goodbye.
Captions 28-29, Ball des Weines: Tombola
Note that even the Auf in Auf Wiedersehen is upper case, but this rather complicated rule is not of great concern, since lower case is also an acceptable form. Remember too that danke can also be the first person singular form of the verb danken, "to thank," and is therefore always written in lower case. Ich danke euch herzlich!