Here some files to reproduce the MySQL-Privileged problem

build

#!/bin/sh

sudo docker build -t mysql-test .

Dockerfile

# MySQL

FROM ubuntu

RUN apt-get update
RUN echo "mysql-server-5.5 mysql-server/root_password password root123" | debconf-set-selections
RUN echo "mysql-server-5.5 mysql-server/root_password_again password root123" | debconf-set-selections
RUN echo "mysql-server-5.5 mysql-server/root_password seen true" | debconf-set-selections
RUN echo "mysql-server-5.5 mysql-server/root_password_again seen true" | debconf-set-selections
RUN apt-get install -y mysql-server mysql-client telnet

steps (version 1)

######################
### Non Privileged ###
######################
sudo docker run -it mysql-test bash

# Inside container
/etc/init.d/mysql start
telnet localhost 3306 # <-WORKS

##################
### Privileged ###
##################
sudo docker run --privileged -it mysql-test bash

# Inside container
/etc/init.d/mysql start

# A lot of '/usr/sbin/mysqld: error while loading shared libraries: libaio.so.1: cannot open shared object file: Permission denied'

steps (version 2)

######################
### Non Privileged ###
######################
sudo docker run -it mysql-test bash

# Inside container
/usr/bin/mysqld_safe &
telnet localhost 3306 # <-WORKS

##################
### Privileged ###
##################
sudo docker run --privileged -it mysql-test bash

# Inside container
/usr/bin/mysqld_safe &
telnet localhost 3306 # <-DOES NOT WORK