Four, I believe (or three as they travel through the fourth, depends on a lot). The rest require post-grad amounts of mathematics.

There is a book (and accompanying video), Imagining The Tenth Dimension that might prove useful, though I’ve been told that it has lost some credibility.

At the present time, I believe it is still four. Time not only counts, but is actually inseparable from the three we perceive purely spatially according to relativity.

String theory, and its descendants, needs additional spatial dimensions in order to work out. Six more (for a total of ten) for the major original string theories and seven more (eleven total) for the still-notional “M theory” that is proposed to connect these theories together along with another separate theory (10-dimensional supergravity). There is a 26-dimensional string theory, but has it only has forces (bosons) and not matter (fermions) as well as faster-than light solutions (tachyons). These extra spatial dimensions are usually posited to be “closed-up” on a very small scale so as not to conflict with the present state of our observations. The frequently used analogy being: a cable seen from a great distance appears one-dimensional, but an ant crawling on it can access the second closed-up dimension on the surface and move around the cable as well as along it.

Normally, it is assumed the scale on which this closing-up happens is on the order of the Planck length, which is 1.616… x 10^-35 meters. This is small. For comparison, a proton is revealed is having internal structure (three quarks swarming around) on the scale of 1×10^-15 meters. As small as protons are to us (fifteen orders of magnitude), the Planck length is much smaller to the proton (twenty orders of magnitude)!

This presents a problem: how can we ever observe these extra dimensions directly if they really do exist? And the answer is never for all practical purposes if they are this tiny. Any experimental verification will be indirect.

I have read that have been some suggestions that extra dimensions may actually exist on a scale much, much larger than the Planck Length – verging on the macroscopic – and yet have eluded our notice. The last I heard, IIRC, was that they combing the data from the Tevatron at Fermilab to come up with some support for the idea, but that was a few years ago now.

There’s nothing to rule out our being embedded in a higher-dimensional spacetime which we cannot access and does not interact with us (or at least not usually). This has been a topic of speculation going back to ancient times. Infinity and the Mind by Rudy Rucker has a short and sweet exposition of the history of these notions. The modern version of this coming out of the development of string theories has our universe existing as a brane floating about within the bulk (see: brane cosmology).

I think that I experience the 5th dimension when I go jogging.

I know where all the mile markers are on my regular jogging course (for the last 15+ yeas) and I remember certain times at each mile marker (slow times, medium times and fast times). When I am jogging slow, I can see in my mind a huge pack of me’s from the past running ahead of me. This pushes me to catch up with the pack. I know when I’m in the back, middle or front of the pack of me’s by the times that I get at each mile split. When I feeling extra strong I run way up in the lead pack and when I look back I can see all of the slower me’s all stretched out behind me.

Weird huh?

@Nullo‘s right as to what we can observe. There are three spatial dimensions and time. @hiphiphopflipflapflop‘s point about time being woven into the fabric of the other three dimensions is an excellent one. Cosmologists generally refer to it as spacetime since only with space and things in it does time exist.

String theory works well with 11 dimensions including time, but if there are these extra dimensions the thinking is that they are tightly coiled and currently unobservable, so even though they are mathematically consistent and actually suggests some solutions to the unanswered cosmological problem of creation, they wouldn’t qualify as observable at this time—unless you have better eyes than mine. :-)

We can’t observe any directly, they arise from the spatial relationships of things to one another. An empty universe would be dimensionless without length, breadth, height or even time.

As of now, three and a half. There are theories that there may be up to eleven, with the rest either massive, with universes floating around in it, or so small as to make them almost non-existant. We have observed three space dimensions, and one time dimension. The time dimension may only be half a dimension, since everything we have seen so far goes the same way in it, with no backwards travel.

@PhiNotPi I’m not quite with you there on the half a dimension. If a train has only a forward gear it doesn’t mean the track has only half a dimension?

You can’t really make that comparison. If the track makes it physically impossible to go backwards, all track behind you might as well not exist. We have no proof that the past exists. All we have is our memories of when it did exist, but does it still exist once we have lived it?

For any who is interested, this thread lead me to search online for more, and I found a PBS miniseries called “The Elegant Universe”. I’ve only just finished Part 2, but I’m posting all three parts here.

For a talk on the multiple-dimensions of the theory, go to Chapter 7 of Part 2.
It also looks like the 3rd Part is completely about multiple dimensions, but I haven’t quite gotten there yet.

They’re a bit long, short of an hour each, but in trying to imagine other dimensions, I like the slower-pace; and the effects they use to explain can be kind of silly, the “Quantum Cafe”, but it’s understandable and kinda fun.

Part 1
Part 2
Part 3

And here’s the PBS homepage of The Elegant Universe (the non-video form), too.

We can think of experiments and be able to observe 10 or 11, while our brains will continue to struggle with this.

There are four. Height, Width, Depth and Spacetime. The latter is hard for many people to get their head around. To put it simply; Mass tells space-time how to curve, and space-time tells mass how to move.

People would argue with me but, as I see it, there are eight.
Height
width
depth
mass
time
velocity
direction
perception