Established, reliable domain name registrars

There are hundreds of accredited domain name registrars. Following link shows complete list of approved domain name registrars.

http://www.icann.org/en/registrars/accredited-list.html

Know more about ICANN:
1. ICANN Web site: http://www.icann.org/
2. Wiki link: http://en.wikipedia.org/wiki/ICANN

Following is the list of well established,  reliable domain name registrars.

Go Daddy 

GoDaddy is the biggest domain name registrar company. They currently have almost 35,654,055 domain names under them, which makes them the most trusted place to buy domain names.

Number of Domain Names: Above 36 million

Enom.com

Enom has been listed among the fastest growing domain name registrars. They have the second largest number of domain names registered under them after GoDaddy.

Number of Domain Names: Above 10 million

TuCows

Tucows is a wholesaler of domain names and other Internet services to ISPs and web hosting companies worldwide. They do not sell domain names directly to consumers. Tucows sells domains through their affiliate sites such as DomainDirect.

Number of Domain Names: Above 7 Million

NetworkSolutions

Network Solutions is the fourth largest ICANN accredited domain name registrar. They offer complete domain name related solutions.

Number of Domain Names: Above 6 Million

Other well known registrars

Namecheap - http://www.namecheap.com/
Register - http://www.register.com/
Moniker - http://moniker.com/
Wild West Domains - http://www.wildwestdomains.com/