Experimental evidence of a fourth spatial dimension? As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist.
Is there any direct or indirect evidence supporting a fourth dimension?
 A: Assuming you mean spatial dimensions, Nima Arkani-Hamed has some ideas and papers on the subject. Here is a popular article:
http://www.nature.com/nature/journal/v433/n7021/full/433010a.html
The idea of a fourth dimension was popularized by Charles Hinton in the late 1800s: http://en.wikipedia.org/wiki/Charles_Howard_Hinton
and, of course, by Edwin Abbott (Flatland).
Then, there are the modern takes on it, like The Higher Dimensions Family Church :)
http://en.wikipedia.org/wiki/Carlton_Pearson
A: In Newtonian physics, there are 3 dimensions of space and there is time. Everything happens in these 3 dimensions of space and time is only a parameter which measures the evolution of things (for example the change in the positions of a particle). 
In Special Relativity, it was postulated that the 3 dimension of space and time should be considered under equal footing and that time should also be seen as a dimension. The fact that that was the case was shown by proving the validity of relativistic transformations between various reference frames which mixed time with space. But because a reference frame in Special Relativity can be considered as a universal frame for that particular observer, it introduced time in an absolute sense and thus one could consider again time to be only a parameter for that particular observer.
General Relativity established the full equivalence of time and space as parts of spacetime, in the sense that it fully geometrised gravity and the place where everything happens is the 4-dimensional spacetime with 3 dimensions of space and 1 of time. The evidence supporting the geometric view of spacetime are those supporting the Einstein equivalence principle which states that every free falling frame doesn't experience any gravitational effects or that an accelerated observer can't distinguish with local experiments his accelerated frame from a gravitational field.
So, to quote some experiments that verify the Einstein equivalence principle, we have:


*

*The redshift of photons climbing the gravitational potential,

*The time delay of clocks in different gravitational fields,

*The Eotvos-type experiments that show that all objects regardless their composition fall with the same acceleration,

*The constancy of physical constants.

A: Not data, but...
The string theorists and other TOE people would say that a single theory exhibiting all the required symmetries only has representations at higher dimensions.
Then they deal "So why can't we see the other ones?" by giving them periodic boundary conditions on length scales smaller than we have been able to probe.
This would be "extra" spacial dimensions, but would not have any interesting effects on day to day life.
A: If we consider the heat capacity of a mono-atomic gas, it will answer the question how many independent degrees of freedom exist in space. Experiment gives three space dimensions.
A: Since you were here yesterday, you're certainly here at the moment and we hope you're here tomorrow, there's some pretty compelling evidence for an additional dimension!
To be less flippant, here is a more vivid example. Suppose an infinitely thin disc (like a coin with no discernible thickness) came into existence on your desk Tuesday midday and remained there until Thursday midday when it vanished. We can think of this object as a cylinder with cross section the disc and "length" two days. Time is being treated here as analogous to the third spatial direction.
To describe a world which extends in time as well as in 3D space we need that extra time coordinate which constitutes a fourth dimension. As other answers indicate, in relativity the time dimension is distinguished from spatial directions by a sign change in the definition of space-time interval.
A: No. Unless you consider time as the fourth dimension. Also current research suggests that spacetime becomes two dimensional at high energies and very small distances. So, yes, dimensionality of spacetime is a mutable concept in modern physics. But no experimental evidence, indirect or direct, has been found for a fourth spacelike or a second timelike direction.

Someone is bound to question my statement that "spacetime becomes two dimensional at high energies and very small distances". My backup reference for this is Carlip's recent paper The Small Scale Structure of Spacetime, where he considers inputs from five or six different theoretical lines of research. Of course, there is bound to be a spectrum of opinion on this issue.
A: If you think of time as we perceive it as a fourth dimension, time is actually all space and vice versa. We can only perceive the present at any time so every moment I guess is its own dimension or universe making time a record of all the space every moment in a way. The past can never be changed and the future can't be affected until it's the present. That's why space time describes it as physically the same but we can only perceive and make use of time to keep track of and keep schedules and memories.
I think it may be more likely that there are infinite time lines and we in our universe or dimension live out one of the infinite combinations of a lifetime of "present moments."  
