What is Ip address 0.0.0.1 for and how to use it?

I don't think you can assign any 0.0.0.0/8 address to a host machine.

The 0.0.0.0/8 network is reserved by IANA as per RCF 5735. It's "Present use" reason is for:

"This" network.

I did find IANA IPv4 Special-Purpose Address Registry page that explains the use of special addresses. In this document it is a bit more specific about 0.0.0.0/8 and says that it's for "This host on this network"

Some important terminology

Source

• A boolean value indicating whether an address from the allocated special-purpose address block is valid when used as the source address of an IP datagram that transits two devices.

Destination

• A boolean value indicating whether an address from the allocated special-purpose address block is valid when used as the destination address of an IP datagram that transits two devices.

Forwardable

• A boolean value indicating whether a router may forward an IP datagram whose destination address is drawn from the allocated special-purpose address block between external interfaces.

Globally Reachable

• A boolean value indicating whether an IP datagram whose destination address is drawn from the allocated special-purpose address block is forwardable beyond a specified administrative domain.

Reserved-by-Protocol

• A boolean value indicating whether the special-purpose address block is reserved by IP, itself. This value is "TRUE" if the RFC that created the special-purpose address block requires all compliant IP implementations to behave in a special way when processing packets either to or from addresses contained by the address block.

0.0.0.0/8 is listed only as a source address and is reserved-by-protocol. It is not a destination, forwardable, or globally reachable address.

After doing a bit more digging, RCF 1122 and RCF 1700 both back this up.

RCF 1122 states a { 0, <Host-number> } address 

Specified host on this network. It MUST NOT be sent, except as a source address as part of an initialization procedure by which the host learns its full IP address.

RCF 1700 states a { 0, <Host-number> } address 

Specified host on this network. Can only be used as a source address.

After all of this looking, I'm still unclear exactly how one of the 0.0.0.0/8 addresses would be used. From my understanding, it seems like one of these addresses are somehow used as a source address, prior to a computer obtaining an IP address.

One guess I have is it could maybe be used in ARP or a DCHP DISCOVERY request. Even with that, in my experience the source address of a DHCP DISCOVERY is 0.0.0.0/32

UPDATE: I found one example where 0.0.0.1 is used in an ARP request for an Alcatel-Lucent line card in which it refers to 0.0.0.1 as the sender of the request.

This could be commonly used in RAPS (Ring Automatic Protection Switching) cards, link aggregation/multi-chassis link aggregation, or other passive hardware.