I know that capacitive touchscreens need two layers of electrodes, but have not found much on the precise shape of those electrodes. Are they just plain straight ribbons, or are there more elaborate shapes around? If yes, what would be the advantage of those shapes?
The ones that I've seen generally use a pattern of diamond-shaped electrodes, like here: An Analysis of Electrode Patterns in Capacitive Touch Screen Panels. I once saw a design for a wall-sized screen that sensed human body movement in front of the wall (some sort of interactive art project) that used the same pattern, but the diamonds were about 4" across.
For phone touch sensing screens the actual shape varies - I have designed them and we would do multiple simulations and subsequent testing of the design to optimize the position sensing linearity.
For phone use they typically have a 5mm pitch with position sensing resolution of about 0.1mm.
There are two main methods of touch sensing - referred to as self-capacitance or mutual capacitance. Small screen devices (watches, small MP3 players) use this technique with diamond shaped electrodes - an electrode per sensing point is required.
Larger screen devices typically use mutual capacitance where a row and column matrix can be used to give many more sensing points but does have some side effects with phantom touches where a touch may be sensed at the wrong place along the same column or row. For these designed the shape may be rectangular.
Some models of the Apple iPhone used electrodes embedded within the LCD display itself and were shared with display functionality - in this case the shape is much more restrictive as the electrodes do double duty.
This "in-cell" sensing is described here: Apple in-cell touch sensing patent
The type of touch screens that use electrical resistance measurements consist of two sheets that each have a uniform layer of a conductive coating across the whole sheet. The two coatings face each other and are held apart a very small amount by tiny glass beads. Touching the screen at any particular point makes a contact between the two layers.
One of the two layers has electrical contact bars that span the full width across the extreme top and bottom of the sheets. The other layer has electrical contact bars that span the full height on the extreme left and right sides. Measuring a touch point takes two steps, one for X and the other for Y. One measurement step follows like this. Tie the two electrical contact bars of one layer together through an analogue switch and connect to the input of an A/D converter. Tie the contact bars of the other layer through analogue switches with one to GND and the other to a reference Voltage level. The touch point acts as a voltage divider across the sheet with the bias across it and the sheet with the bars tied together acts as the center sense point of the voltage divider. The A/D reading corresponds proportional to the touch point distance from one of the conductive bar to the other. Repeat process by reversing the layer roles to get the other direction.