Googling info on this is quite hard. Just to share what is the procedure. Basically you need to apply for a new Singpass based on your new PR NRIC, as the previous Singpass based on FIN will be invalid.
Also, most would soon encounter this problem: Your mobile phone is tied to the old Singpass account, and the system would not let you use it for the 2FA.
The solution (took quite a while for me to find this): (Source)
Q: I am a FIN holder who has been granted Singapore PR, how can I continue to use SingPass?
Ans: You will need to register for a new account using your NRIC number. Upon successful registration, your account details will be mailed to you within four working days. Click here for more details. Please note that one mobile number can only be registered with one SingPass account. If you have previously registered your mobile number in your existing account, and would like to use the same number for your new account, please email firstname.lastname@example.org with the following details:
Your FIN number
Your new NRIC number
Your mobile number
Source: Singpass FAQ
Disclaimer: This info is dated 21 October 2017. The procedure may or may not change in the future. Also, please double check with the official website before taking any actions.