Visa status after quitting my job and before graduate school?

Can I legally remain a tenant and rent my HDB unit in between my S pass and student visa approval?

Probably not officially. But you could try to talk to your landlord and find a mutually agreeable solution.

Do I need to leave SG and return after my LTVP expires to be under the 90 day tourist visa?

You must leave Singapore before your LTVP expires.

It is uncertain wether a tourist visa is issued when you return before your student pass is appoved. It is better not to test this, as you might be denied entry - and that could affect your student pass application.

Are there any other issues I should look out for?

Do not try to find ways around the rules (and their common interpretation by the authorities). This will always backfire in Singapore!

@beppi

Hi! Thank you for your reply, makes sense everything you shared, definitely don't want to risk my long term future in SG.

With regards to the tourist visa. As I am from a country that is visa exempt (90 day tourist visa), does that still mean I can be denied entry into SG?

The immigration officers can deny entry at any time for any reason they like.

They rarely do it without good reason, but a short absence after a non-renewable pass expired (which looks like a visa-run) is definitely a No-Go!