O host local é normalmente 127.0.0.1. Todos os endereços 127.x.x.x são considerados endereços de loopback. Eu acho que o Firefox é a exceção neste caso.
"Loopback" IP addresses:
127.0.0.0 - 127.255.255.255
Each computer on the Internet uses 127.0.0.0/8 to identify itself, to itself. 127.0.0.0 to 127.255.255.255 is earmarked for what is called "loopback". This construct allows a computer to establish/validate its IP stack. Most software only uses 127.0.0.1 for loopback purposes (the other addresses in this range are seldom used). All of the addresses within the loopback address are treated with the same levels of restriction in Internet routing, so it is difficult to use any other addresses within this block for anything other than node specific applications, generally bootstraping. This is documented in RFC 3330.
(extraído de IANA.org)
O endereço 0.0.0.0 é:
0.0.0.0/8 reserved for self-identification [RFC5735]
(tirada novamente da IANA)
e em detalhes:
0.0.0.0/8 - Addresses in this block refer to source hosts on "this" network. Address 0.0.0.0/32 may be used as a source address for this host on this network; other addresses within 0.0.0.0/8 may be used to refer to specified hosts on this network ([RFC1122], Section 3.2.1.3).
(retirado do RFC 5735)