GoDaddy

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Based on my experience with a shared hosting account on GoDaddy in 2012:

Their ssh access is weird. You can only activate it by receiving a PIN via phone, and that only allows the master account to use ssh -- which means that any outside developers you have doing work for you may need to know the credentials of your main account.
Their server does not have rsync installed.
Their standard FTP access sticks you inside the html folder, with no access to areas outside of it. This makes certain operations more difficult and prevents using best security practices for certain things, e.g. .htpasswd files.
Their MySQL access is also weird. Given an account with "unlimited databases", you can only create one database per instance -- where each instance has a different domain name and username/password. This means that you cannot do queries between databases, nor can you use phpMyAdmin to directly clone tables from one database to another; the best you can do is export from one db and import to the other. The benefits to GoDaddy of this scheme are unclear at best.

2011: GoDaddy came out in support of some particularly evil legislation (SOPA/PIPA); there has been much discussion...
2007-03-11 Godaddy DOWN (discussion thread)
2007-01-25 GoDaddy pulls security site after MySpace complaints: GoDaddy suspended a domain of one of their registrants with almost no warning
2006-05-10 GoDaddy has begun to suck