Python, 77 46 44 bytes

lambda s:[a*s.count(a)for a in set(s)-{' '}]

Pretty simple. Goes through the unique characters in the string by converting it to a set, then uses a list comprehension to construct the exploded diagrams by counting the number of times each character occurs in the string with str.count. We filter out spaces by removing them from the set.

The order of the output may vary from run to run; sets are unordered, so the order in which their items are processed, and thus this answer outputs, cannot be guaranteed.

This is a lambda expression; to use it, prefix lambda with f=.

Try it on Ideone! (Now with all test cases!)

Usage example:

>>> f=lambda s:[a*s.count(a)for a in set(s)-{' '}]
>>> f('Ah, abracadabra!')
[',', 'A', 'aaaaa', 'd', '!', 'bb', 'h', 'c', 'rr']

Python 3.5+, 77 46 44 41 bytes

lambda s:[a*s.count(a)for a in{*s}-{' '}]

Pretty simple. Goes through the unique characters in the string by converting it to a set (using Python 3.5's extended iterable unpacking), then uses a list comprehension to construct the exploded diagrams by counting the number of times each character occurs in the string with str.count. We filter out spaces by removing them from the set.

The order of the output may vary from run to run; sets are unordered, so the order in which their items are processed, and thus this answer outputs, cannot be guaranteed.

This is a lambda expression; to use it, prefix lambda with f=.

Try it on Ideone! Ideone uses Python 3.4, which isn't sufficient.

Usage example:

>>> f=lambda s:[a*s.count(a)for a in{*s}-{' '}]
>>> f('Ah, abracadabra!')
[',', 'A', 'aaaaa', 'd', '!', 'bb', 'h', 'c', 'rr']

Saved 3 bytes thanks to @shooqie!

Python, 77 46 44 bytes

lambda s:[a*s.count(a)for a in set(s)-{' '}]

Pretty simple. Goes through the unique characters in the string by converting it to a set, then uses a list comprehension to construct the exploded diagrams by counting the number of times each character occurs in the string with str.count. We filter out spaces by removing them from the set.

This is a lambda expression; to use it, prefix lambda with f=.

Python, 77 46 44 bytes

lambda s:[a*s.count(a)for a in set(s)-{' '}]

Pretty simple. Goes through the unique characters in the string by converting it to a set, then uses a list comprehension to construct the exploded diagrams by counting the number of times each character occurs in the string with str.count. We filter out spaces by removing them from the set.

The order of the output may vary from run to run; sets are unordered, so the order in which their items are processed, and thus this answer outputs, cannot be guaranteed.

This is a lambda expression; to use it, prefix lambda with f=.

Try it on Ideone! (Now with all test cases!)

Usage example:

>>> f=lambda s:[a*s.count(a)for a in set(s)-{' '}]
>>> f('Ah, abracadabra!')
[',', 'A', 'aaaaa', 'd', '!', 'bb', 'h', 'c', 'rr']

Python, 77 bytes

from collections import*
lambda s:[a*b for a,b in Counter(s).items()if' '!=a]

Pretty simple. Uses collections.Counter, Python's version of a multiset/bag, to count each thing in the string, then uses a list comprehension to filter out the spaces and create the exploded views of each character.

This is a lambda expression; to use it, prefix lambda with f=.

Python, 77 46 44 bytes

lambda s:[a*s.count(a)for a in set(s)-{' '}]

Pretty simple. Goes through the unique characters in the string by converting it to a set, then uses a list comprehension to construct the exploded diagrams by counting the number of times each character occurs in the string with str.count. We filter out spaces by removing them from the set.

This is a lambda expression; to use it, prefix lambda with f=.