I think what you are looking for is how to get the ISP's default route to advertise it into MPLS network, correct? then your question if that route should go to a P, PE o ASBR refers to the type of connectivity to establish with Internet Gateway Router.
AS everything in routing, it depends.
If you want to handle the default route independent from your MPLS infrastructure, then the PE is the option, you will have the default route into a VRF and with a RD you import this route from any other PE, providing customers with internet, however, you could have some issues when addressing from multiple customers overlap.
You can also get the default route into your MPLS infraestructure (global VRF) and do some import-map to get the route into customer's VRF at a PE router, like having an extra customer's site where they "get" internet, this could happen in ASBR, but importing the route happens always with BGP.
For the P router option, it's similar, you get the route into BGP and then you can import it into multiple VRF always in a PE, this is not that good because customer's traffic will traverse global vrf for your network, and ultimate goal to have L3VPN is to isolate multi tenant traffic.
For advice you can read MPLS fundamentals, chapter 7 MPLS VPN, section Internet Access page 237, there you can find how to advertise a default route in an MPLS network. I had the honor to get MPLS training from Luc, the author, he's a great guy and has a deep understanding of ISP architecture.
Please @Ibrahim Jamil if you consider an answer helps you, mark it as useful.
Edit: I found some useful links:
Internet Access from an MPLS VPN Using a Global Routing Table
Introduction to Cisco MPLS VPN Technology
https://www.cisco.com/en/US/tech/tk436/tk428/technologies_white_paper09186a00800a85c5.shtml#wp31356
https://www.cisco.com/en/US/tech/tk436/tk428/technologies_white_paper09186a00801281f1.shtml#wp39724
CCIE 52804
