Do I need to apply copyright for my graphic illustrations and designs before posting on social media and art freelance websites?

There is no need to apply or file for registration to get copyright protection, and IPOS does not administer any process for registering copyright in Singapore. Copyright protection automatically arises if the conditions under the Copyright Act 2021 are met.
The relevant conditions are:
The creator must have a connection with Singapore (e.g., by being a Singapore citizen or resident when creating the work, or by first publishing the work in Singapore).
For literary, dramatic, musical or artistic works (including graphic illustrations and designs), the expression of the work must be original and in a tangible form, such as in a recording or writing. Originality simply means that there is a degree of independent effort in the creation of the work.
You may refer to the Copyright Infopack for more details on the above.