Difference between 沒轍, 莫可奈何，沒辦法
These three phrases differ in register, which affects how and when they will be used.

méi bànfǎ 沒辦法 is basic Mandarin and can be used in ordinary speech and writing without any issues arising.

mò kě nàihé 莫可奈何 is a highly learnèd idiom; a more common form is wú kě nàihé  無可奈何. As four-character Classical idioms, they have a certain formal "air" to them, even though 無可奈何 is pretty common. But it is a less basic expression than 沒辦法 — I would use 沒辦法 nine times out of ten, at least. I have never used 莫可奈何.

méizhé 沒轍 is highly colloquial and has a regional flavor. I wouldn't use it unless it's part of your normal regional accent.