Five years or so ago you would be hard pushed to find reliable SSH for your shared hosting environment. However as web development has grown in popularity and the barrier to entry has lowered, SSH is becoming a more common offering.
Firstly what is SSH? Secure Shell, a secure encrypted command line interface which can be accessed remotely (normally.)
Now SSH offers a lot more freedom and flexibility in a standard server (assuming you are running a Linux distribution.) The idea behind using only SSH is you have no need to consume resources for a GUI when performing basic tasks. You can install everything you need and get a web server up and running using just SSH.
Now the major advantage to SSH is the complete and true flexibility to do whatever you need. However on a shared server they heavily restrict what you can and cannot do with SSH.
Now if SSH is part of the offering for your normal web service then you can start using it no problems. However if you are looking to change or buy new web hosting then should you be looking for SSH?
There are a few questions you need to ask yourself. Firstly do you require a service which does not have a good GUI? In which case SSH may have to be for you. A prime example would be using GIT on your server. If you are looking to install custom programs/software then you cannot use SSH on a shared server.
If you only need the bare basics of SSH and you are not paying extra for the SSH then by all means go for it. However if you are paying extra and do not require any command line features, you might be better off looking for cheaper more basic web hosting.
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