[Harri (OP)]

I was intrigued to read the universe being described as 'flat'. To what degree is this accurate?

If the universe is flat then is it possible to detect a direction in which the expansion is travelling? Perhaps taking the earth/solar system as a point of reference.

[Kryptid]

It's expanding in all directions at once.

[Halc]

If the universe is flat then is it possible to detect a direction in which the expansion is travelling? Perhaps taking the earth/solar system as a point of reference.

From any spatial point of reference, like here, the universe spatially expands equally in every direction.

From an objective viewpoint, I suppose the balloon analogy is appropriate.  Earth is a dot on the balloon, and all the other stuff is other dots drawn elsewhere.  As the balloon expands equally, every dot moves away from a given reference dot in proportion to its distance from that reference dot.  This is exactly as we observe things.

So is there a direction to the balloon expansion?  Yes, but it is outward, not in any spatial direction that you can point with an arrow on the balloon surface.  In the same way, there is no arrow you can make showing the expansion direction.  The direction is outward in that 4th direction that you can't point with a spatial arrow.

[evan_au]

The Solar System (and our local galaxy cluster) seems to have a measurable motion with respect to the Cosmic Background Radiation. The CMBR seems to be slightly red-shifted in one direction, and slightly blue-shifted in the opposite direction.

Whether this represents random motion of our galaxy, a local gravitational concentration or a random temperature variation in the CMBR is not known for certain.

See: https://en.wikipedia.org/wiki/Cosmic_microwave_background#CMBR_dipole_anisotropy

[chiralSPO]

Is there a detectable direction in which the universe is expanding?

Outwards. 

[Harri (OP)]

So expanding equally in all directions at the same time. If the universe was 'flat' then I assumed that there must be an observational spatial direction in order to determine it was flat.

[jeffreyH]

The Solar System (and our local galaxy cluster) seems to have a measurable motion with respect to the Cosmic Background Radiation. The CMBR seems to be slightly red-shifted in one direction, and slightly blue-shifted in the opposite direction.

Whether this represents random motion of our galaxy, a local gravitational concentration or a random temperature variation in the CMBR is not known for certain.

See: https://en.wikipedia.org/wiki/Cosmic_microwave_background#CMBR_dipole_anisotropy

I'll have to study that if I ever get the time.

[Halc]

If the universe was 'flat' then I assumed that there must be an observational spatial direction in order to determine it was flat.

Flat means that parallel lines stay the same distance apart, in any direction.  Yes, direction matters, but our space seems symmetrical this way.  With positive curvature (like the balloon), parallel lines meet.  With negative curvature (like kale), parallel lines diverge.

[Harri (OP)]

Nicely cleared up.