In this sentence why isn't "der Couch" in the accusative or dative i.e. den Couch or dem Couch? I thought after the word "auf" you always use accusative or dative.
Du kannst mein Bett haben, und ich schlafe auf der Couch.
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Your assumption about auf taking the dative or accusative case is absolutely correct. The place where you have gone wrong is in thinking that Couch is necessarily a masculine noun. According to Duden, Couch is a masculine or feminine noun, and the sentence does indeed have Couch in the feminine dative.
In your sentence, it is the preposition auf that determines the case of Couch.
"auf" can rule either the accusative (with a directional movement) or dative (without). There's no directional movement involved, so Couch needs to be (and actually, is, in your example, as "die Couch" is feminine) in dative.
Couch being masculine (opposed to feminine everywhere else where German is spoken) is actually a (nearly) neglectable exception valid for Swiss German only - which is, in fact, neglected by many dictionaries. Your dictionary, if it's a good one, should make that clear.