How can I tell if a library was compiled with -g?

If you're running on Linux, use objdump --debugging. There should be an entry for each object file in the library. For object files without debugging symbols, you'll see something like:

objdump --debugging libvoidincr.a
In archive libvoidincr.a:

voidincr.o:     file format elf64-x86-64

If there are debugging symbols, the output will be much more verbose.

The suggested command

objdump --debugging libinspected.a
objdump --debugging libinspected.so

gives me always the same result at least on Ubuntu/Linaro 4.5.2:

libinspected.a:     file format elf64-x86-64
libinspected.so:     file format elf64-x86-64

no matter whether the archive/shared library was built with or without -g option

What really helped me to determine whether -g was used is readelf tool:

readelf --debug-dump=decodedline libinspected.so

or

readelf --debug-dump=line libinspected.so

This will print out set of lines consisting of source filename, line number and address if such debug info is included into library, otherwise it'll print nothing.

You may pass whatever value you'll find necessary for --debug-dump option instead of decodedline.

What helped is:

gdb mylib.so

It prints when debug symbols are not found:

Reading symbols from mylib.so...(no debugging symbols found)...done.

Or when found:

Reading symbols from mylib.so...done.

None of earlier answers were giving meaningful results for me: libs without debug symbols were giving lots of output, etc.