Hi Fraser, Pilou has given you some ideas and I'll give a couple of more tips here.

One problem that is making lofting or sweeping these things difficult is that a couple of your curves are made out of more than one segment. I selected them and did Edit/Separate, and found these segments:



This is especially not good because the second one has kind of a crease in it, this will make it more difficult for it to blend nicely with the next one up. So to fix this, I selected those segments and the adjacent ones, then did Edit/Join, then selected the new joined curve and did Edit/Show points, and selected the point where they join together and deleted it. This is what it looks like just before deleting those juncture points:



After doing that, those segments are fused together into a single segment, and the crease in the curve is removed.

There is kind of a similar problem with this curve:



It is also made up of different segments, and one of the join points is not smooth.

Each time you have a curve like this that is not a smooth piece, it will introduce alignment and further segmentation problems when you try to surface it.

So for this one you can smooth it out in the same way as above, or if you want it to be creased, you can dice up the upper curve into segments as well by using Edit/Trim. Here I have done that to this curve:



So that it is now in 3 segments as well.

Now that you have things matching up with smooth pieces, it is possible to sweep them into a surface - here is the start of that - start by selecting these 2 curves:



Now do a Construct/Sweep, and select the 2 adjacent longer curves as the rails. That will give you this result:



So you can repeat that to surface the rest of the shape. This piece is made up of nice smooth parts in both directions, so it can be swept all in one go:



to get this result:



So you've got a ok framework there - the main problem that you've got is a few creases and segments in bad places which prevent getting good results from trying to use larger sweeps.

Let me know if you need more information about any one of these steps.

- Michael