Make a selection first—Copy Command+C and Cut Command+X operate on the selected pixels and are disabled without a selection. Copied pixels land on the clipboard as a regular image, so they paste into any app. In Composite the copy is the stacked visible layers inside the selection. In Solo, Onion Skin, and Animation it is the selected layer only, so neighbouring and other on-screen layers are left out. In Tileset mode the copy is taken from the sheet as you see it, not from tiles stacked on top of each other.
Choose Edit > Copy Merged Shift+Command+C to copy the entire canvas as it is composited from all visible layers, including layer opacity and transparency. Copy Merged does not require a selection. In Tileset mode that composite is the sheet (or the Wrap Tiles group), matching Export as Image.
Choose Edit > Paste Command+V. The image arrives as a floating selection you can position before setting it down, and the layer it lands on keeps its name. Dottie pastes:
Choose Layer > New Layer with Clipboard instead to create a new pixel layer directly from the clipboard's bitmap image, fitted and centred to the canvas—unlike Paste, it lands as a regular layer rather than a floating selection. Choose File > New from Clipboard Option+Command+N when the clipboard should become its own document; that uses Import, so the canvas is sized to the image.
Choose Edit > Copy Cursor Position Control+Option+Command+C to copy the hovered pixel’s coordinates as “x,y” text—useful when writing scripts or documenting sprite offsets.