Electrode specification
The exact choice of electrode really depends on the goal of the experiment and what will be in your cell:

The most basic requirement is that the solvent or electrode material not decompose at the potential you want to work at.
For each solvent/electrode combination, there is a potential window in which you can safely work.
In the most common situation—working in aqueous solutions, the choice of working electrode mainly depends on whether you want to observe oxidation or reduction.
Electrodes like platinum or gold are commonly used for oxidation reductions because they have high overpotential for oxygen evolution in aqueous solutions (platinum in particular, performs poorly for reduction as it is rather good at catalysing the hydrogen evolution process).
Conversely, for reduction reactions, mercury and bismuth have potential windows that extend to quite negative potentials, but will themselves oxidize as you move towards positive potentials.
There are also some materials that work pretty well for both positive and negative potentials, such as glassy carbon and boron-doped diamond.

For analytical purposes, you can get away with (and often prefer) small electrodes, so cost is less of a concern, but if the application is mass electrolysis, large electrodes are required to have a decent reaction throughput and more expensive materials may be impractical.
Cheap materials like stainless steel and graphite can be used as electrodes in some cases, though they are more limited in the systems in which they may be used.

As far as the reason to choose platinum over lead oxide, gold, or silver is concerned, the standard electrode potential isn't a primary consideration. (Silver's is not higher than platinum's, incidentally)
The standard electrode potential reflects the thermodynamic situation of a particular half-reaction, but is not terribly informative of how the material will behave in a given cell as it does not account for the kinetics of the reaction or any other factors.
Silver is usually not a good choice for a working electrode as it oxidizes quite easily in aqueous solutions, gold is actually a fairly common choice as it has a wider potential window than platinum in aqueous solutions, but it is easily fouled by organic/biological contaminants.
Platinum doesn't have the widest potential window, but is generally resistant to fouling and being reacted itself (giving it a long lifetime).

In a nutshell, electrode choice is dictated by many competing factors: cost, lifetime, potential range, mechanical considerations, etc.
In an analytical lab, screen-printed gold electrodes are quite common because they are relatively inexpensive, suitable for many different reactions, and can be discarded and replaced when fouled.
In an aluminum refinery, you'll find carbon and coke electrodes used in the Hall-Héroult process, as they need to be very large and withstand the high temperatures of molten cryolite.
In the cloralkali process, the anode is usually made from titanium as it must resist the highly oxidizing conditions that produce chlorine, but the cathode can be made from much cheaper nickel as that half of the cell is only used to reduce water to hydroxide and hydrogen gas.