Transistors failed to stop current
Obviously the other guys beat the safety issue to death, so I won't repeat it here other than to say that they are absolutely correct.  

What I want to focus on is why a MOSFET won't work.  Even a MOSFET with proper voltage and current ratings.  Take a look at the first page of the datasheet for the MOSFET you selected.  There is a drawing showing the symbol for a MOSFET inside of the SOT-25 package.  If you look closely, there is also a diode between the source and the drain.

That diode goes by several names:  parasitic diode, body diode, etc.  The diode is not a separate part of the MOSFET that someone put in there, it is a side effect of how a MOSFET operates.  All MOSFET's have them, both N and P channel MOSFETs.  The diode is not always shown in the diagrams and schematics, but it is always in the real device.

That diode is why MOSFETs are not suitable for switching AC signals.  Even if the "switch" part of the MOSFET is turned off, the diode will conduct when the voltage is reversed.  In the case of your bulb, if the MOSFET is turned "off" the bulb will still be on for 50% of the AC cycle since the diode will be conducting.

That diode is not a very good diode.  It is not fast, and has limited current handling ability.  Sometimes you will see a MOSFET with a separate diode in parallel.  This is done for circuits where that parasitic diode is wanted, but it doesn't work well enough.  You normally see this in high current switching applications like power supplies and motor control.

There is a another reason why your circuit won't work, aside from the voltage spec of the MOSFET and the safety stuff.

It is likely that the +5v power supply for the Ardunio is isolated from the AC mains.  This is done for the same safety reasons that everyone was yelling at you for.  But the effect of that  is that the Arduino is floating with respect to the AC mains (and the source and drain of the MOSFET).  And that means that the Arduino cannot generate the proper Gate To Source voltage to actually switch on the MOSFET.  

The "Fix" for this is to connect the source pin of the MOSFET to the GND of the Arduino.  BUT DON'T DO THAT UNLESS YOU HAVE A DEATH WISH!  I only mention this so you can better understand what's going on.  If you really did connect this would would have certainly blown up your Arduino and possibly killed yourself. 

Now that I've given you a hopefully useful answer, I must address the non-technical part of this Q&amp;A.  It is unfortunate that this Q&amp;A quickly degenerated into a semi-rude shouting match.  People on both sides could have acted in a more civil way.  But you have to understand this:  what you were doing could have killed you, started a fire, or done other bad things and it was super important that you stopped doing it as quickly as possible.  In cases like this a certain amount of rude yelling is understandable.  Not ideal, but understandable.  Every parent has yelled very loudly when they see their toddler climbing onto the stove or doing something equally dangerous.  But this yelling due to imminent danger doesn't really translate well in a forum like this.

Sorry, I don't mean to compare you to a toddler!  The point is this:  Sorry for the shouting, but the people here only had your best interest in mind.

There are very few things that will get an Electrical Engineer's blood pumping faster than to see someone inexperienced hooking things up directly to the AC mains.  When you do that things can very quickly degenerate into something very tragic.  Like the time I decided I could juggle knives when I could barely juggle balls.  It ended badly.  True story!