The Pen Tool is the most accurate way to make a selection but it must be done by hand, tracing around the part of the image to cut out.
It produces a vector mask with points and curves, all of which can be edited.
The vector path can be turned into a selection by right clicking inside the mask with the Pen Tool and selecting Make Selection, which brings up a dialogue box. It is best to check Anti-Aliased to smooth the edges slightly and Feather Radius of something like 1 pixel to further add a little smoothness to the edge.
P selects Pen Tool.
CMD + CLICK - Brings back the path, and the points on it, select the Pen tool, select the Path if necessary.
CMD + CLICK & DRAG edits a point or a control arm.
Holding ALT converts the Pen Tool to the Convert Point Tool.
ALT + CLICK on a curve point changes the point to be an angle rather than a curve, then use the handle to show the corner shape.
ALT + CLICK & DRAG on an angle point converts it to a curve.
Hovering over a point converts the Pen Tool to a Delete Anchor Point Tool.
Hovering over part of a curve converts the Pen Tool to a Add Anchor Point Tool.
The vector shape shows up in Paths, with Work Path as the one you are currently creating. Save this by double clicking it in Paths and giving it a name.
You can also create a new Path by clicking on the icon at the bottom of the Paths panel, which calles it Path 1 etc, which saves automatically.
Clicking from one point to another produces straight lines between points.
Clicking and dragging produces a curve between points.
You can produce a stroke along a path using several tools, such as the brush tool.
Pen Icon - Can have other pens included. Peanut Dash is there by default.
Path - Can also have Shape or Pixels.
Selection - Creates a Selection on the current Layer.
Mask - Creates a new Layer with its own Vector Mask. Or clicking this with the Path active creates a Mask with that vector path as a selection.
Shape - Creates a filled shape with a defined stroke for an outline.
Sample Size - Can be Point Sample or a larger size. It opens this dialogue box -
The Properties Panel also displays further options.
Paths can be added to, subtracted from or aligned, and changed stacking order.
Rubber Band Function is turned on and off from the cog wheel, which makes the pen path predictive.