Third rail is used on "heavy rail" (subways like BART, Washington
Metro, etc.) operations largely because, combined with lower
profile equipment, it allows considerable savings on tunnel
construction by using smaller diameter tunnels.

Third rail works best on a system that has its own isolated
right of way, with no grade crossings.

Third rail gets really complicated on locomotive-powered
lines at grade crossings and complex trackwork. With a
single locomotive, you have at best two pickup points --
and a slow-moving train could theoretically get stranded
at locations where there is no third rail contact. Yes,
in practice, you coast through those sections.

As others have already pointed out, there are voltage
limitations, and most third-rail systems use somewhere
around 600-700 volts DC. That requires a lot of substations
to feed the line. It also puts some limitations on how much
power a single locomotive/MU can draw.

Higher voltage AC not only provides more power but also
carries much further between substations.

While catenary can get quite complex, you can still have
continuous coverage over even the most complicated trackwork.

Most European railroads are electrified with overhead
catenary. I think the UK still has some main line operations
that use third rail, but on the continent, only subway
systems use third rail, as far as I know.

European railroads have become quite adept at catenary
maintenance uner all kinds of conditions, and if/when it
becomes damaged, it is usually repaired fairly quickly,
unless the support masts themselves are damaged or
destroyed in a derailment. And, even then, the catenary
is often repaired in less than a day. (Obviously, the
railroads have stocks of spare catenary supports, and
the masts are designed to shear off from their concrete
base on impact.

Most electric locomotives have at least two pantographs.
(Some European locomotives have more, due to different
clearance and other specifications of different railroad
systems.) The normal practice is to run with the rear
pantograph up, so that if it snags an obstacle and
becomes damaged, the debris will not affect the other
pantograph, which can still be raised to get the train
to the next maintenance base.

With two electric locomotives at the front of a train,
the normal practice (at least in Europe) is to run with
the front pantograph up on the first engine and the
rear pantograph on the second engine. This increases
the distance between the contact points of the two
pantographs and reduces the impact that oscillations
in the wire caused by the first pantograph have on
the second pantograph.

There are some special situations where a single
locomotive runs with the front pantograph up. These
include any type of car and/or cargo right behind
the locomotive that would be adversely affected by
the small amount of soot/debris that is caused by
the occasional arcing between the pantograph and wire.

-- Ernest