In the Batman movies, Batman is called when a powerful yellow spotlight with the bat drawing is projected in Gotham City's clouds. The unique drawing then become very clearly visible in the clouds on the entire city. Is this actually possible or at least plausible?
YES, it is possible to project images onto clouds, and it has been for quite some time.
A more recent example:
From The New York Times (2008)
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The key word is definitely "clouds".
In theatres, we do what is similar to the bat signal all of the time, with something called "gobos". Of course, gobos are typically pointed at something solid, which makes them work.
Pointed a gobo at a cloud would work, altough not as well, as clouds are not solid.
Now, if the bat signal needed to be used on a clear night... I am guessing that wouldn't work as well (or at all), since it wouldn't have anything to reflect back on.
Of course, having any flood light, using a special color would still probably grab the attention of the Dark Knight.