How do I perform a case insensitive search?

Replace -name option with -iname as follows:

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find / -type d -iname "apt" -ls

OR

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find / -type d -iname "apt"

The patterns ‘apt’ match the directory names ‘apt’, ‘APT’, ‘Apt’, ‘apT’, etc.

How do I find a directory called project.images?

Type any one of the following command:

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find / -type d -iname "project.images" -ls

OR

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find / -type d -name "project.images" -ls

OR

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find / -type d -name "project.images"

A note about locate command

To search for a file/dir named exactly project.images (not *project.images*), type:

locate -b '\project.images'

find and delete all file in ./ that haven’t been modified since 90 day:

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find ./ -mtime 90 -exec rm -Rf {} \;

I'd strongly suggest not to use find -L for the task (see below for explanation). Here are some other ways to do this:

If you want to use a "pure find" method, it should rather look like this:

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find . -xtype l

(xtype is a test performed on a dereferenced link) This may not be available in all versions of find, though. But there are other options as well;

You can also exec test -e from within the find command:

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find . -type l -! -exec test -e {} \; -print

Even some grep trick could be better (i.e. safer) than find -L, but not exactly such as presented in the question (which greps in entire output lines, including filenames):

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find . -type l -exec sh -c "file -b {} | grep -q ^broken" \; -print