The singer of Oasis sings with style.
Der Sänger von Oasis singt mit Stil.
Is mit Stil correct in this context? Can I also use stilisch?
"mit Stil" or "sehr stilvoll" are both possible. I'm not sure though if in English the phrase "with style" also has a strong connotation to elegant, as it does in German.
"stilisch" is not a proper German word. You either mean the English word "stylish" that is used a lot by young Germans or the word "stilistisch", which would translate to "stylistic".