I have installed shiny server on Ubuntu 14.04 LTS running under a VM. This post is a collection of commands that I had used for setup. Note that if you want to use shiny server on LAN then use Bridged Network Adapter.
On VM you can access the shiny server via localhost:3838 and on remote machines, you can use IP-address:3838 to open it. You can host your shiny app in the /srv/shiny-server/ directory. You can go to /etc/shiny-server/shiny-server.conf to change default configurations like you can change port with listen <port number>; in server {} module.
$ sudo sh -c 'echo "deb http://cran.rstudio.com/bin/linux/ubuntu trusty/" >> /etc/apt/sources.list'
$ sudo gpg --keyserver keyserver.ubuntu.com --recv-key E084DAB9
$ sudo gpg -a --export E084DAB9 | sudo apt-key add -
$ sudo apt-get update
$ sudo apt-get -y install r-base
$ sudo su - -c "R -e \"install.packages(c('shiny','shinyjs','shinydashboard','shinythemes','devtools','rmarkdown'), repos = 'http://cran.rstudio.com/')\""
$ sudo apt-get install gdebi-core
$ sudo wget https://download3.rstudio.org/ubuntu-12.04/x86_64/shiny-server-1.4.2.786-amd64.deb
$ sudo gdebi shiny-server-1.4.2.786-amd64.deb
$ sudo start shiny-server
$ sudo status shiny-server
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