Capistrano 2 brings the shell command which allows you to run commands on your deployment targets.
There is also invoke to run a command directly from your terminal.

Both commands allow running Capistrano tasks or shell commands, and scope to individual machines or machine roles.

Unfortunately Capistrano 3 does not include these commands any more.

cap shell

Basics

First of all, spawn a Capistrano shell (we're using the multistage extension here):

Copy$ cap staging shell

In your "cap" shell you can now run Capistrano tasks by prepending an exclamation mark. For example:

Copycap> !deploy:revision
 ** [out :: web1.example.com] 9aeed9030db6a7b5b4ba077337545677b4fd5c22
 ** [out :: web2.example.com] 9aeed9030db6a7b5b4ba077337545677b4fd5c22
 ** [out :: queue.example.com] 9aeed9030db6a7b5b4ba077337545677b4fd5c22

(deploy:revision is a custom task to check the deployed revision)

To run shell commands on each machine, simply put them there as if you were on a regular terminal session:

Copycap> uname -o
 ** [out :: web1.example.com] GNU/Linux
 ** [out :: web2.example.com] GNU/Linux
 ** [out :: queue.example.com] GNU/Linux

Running more than one shell command

Be aware that shell commands will be sent once, and that your shell on the remote machine is not persistent:

Copycap> cd /tmp
cap> pwd
 ** [out :: web1.example.com] /home/deploy
 ** [out :: web2.example.com] /home/deploy
 ** [out :: queue.example.com] /home/deploy

Instead, queue them:

Copycap> cd /tmp && pwd
 ** [out :: web1.example.com] /tmp
 ** [out :: web2.example.com] /tmp
 ** [out :: queue.example.com] /tmp

Running commands only on individual machines

You may have some commands that you want to run only on a subset of machines, e.g. machines running queue workers.

To pick machines by name, use the on keyword:

Copycap> on web1.example.com uname -o
 ** [out :: web1.example.com] GNU/Linux

Join multiple machine names with commas (e.g. web1.example.com,web2.example.com).

The above is a bit impractical (you could just SSH to a single machine), but you can also scope to entire role groups using the with keyword:

Copycap> with web uname -o
 ** [out :: web1.example.com] GNU/Linux
 ** [out :: web2.example.com] GNU/Linux

^
cap> with queue uname -o
** [out :: queue.example.com] GNU/Linux

For more information, see the linked page.

cap invoke

To run shell commands on all servers, you can also use Capistrano's invoke. Its syntax is a bit more complicated:

Copy$ cap staging invoke COMMAND='uname -o'
 ** [out :: web1.example.com] GNU/Linux
 ** [out :: web2.example.com] GNU/Linux
 ** [out :: queue.example.com] GNU/Linux

Note the COMMAND environment variable which must hold the command(s) you want to run.
As mentioned above, if you want to run multiple commands, you need to chain them:

Copy$ cap staging invoke COMMAND='cd /tmp && pwd'
 ** [out :: web1.example.com] /tmp
 ** [out :: web2.example.com] /tmp
 ** [out :: queue.example.com] /tmp

Note that you probably should not use these commands for complex shell operations.
Instead, maybe create a file that does things for you and run that one instead. Or try an application like Terminator Show archive.org snapshot that allows sending the same keystrokes to multiple machines.