No ships can be modified. However "modified" or "Variant" versions of ships may become producible. In the future with the release of Research and Development read rules here you should be able to make a slightly different version of the ships for which you have Data cards. If you dont own any data cards you will probably have to research and unlock them.
They are calling these "Variants"
You will not be able to increase your personal Bff-Bulk speed from hyper 4 to 15, but just guessing perhaps with engine research you might be able to produce a ship datacard that is a speed or two above standard and will be able to produce BFF-Bulks with hyper 5, or perhaps sublight of 40 instead of 20. Or maybe you put research into weapons and you are able to make a Variant BFF-Bulk datacard with two turbolasers, for whatever reason. I still consider this modifying ships, but not in the way that a lot of new players care about. It will likely take a good deal of time and resources to gather enough research points to make Variants anyway.
Nothing major, to keep the game balanced.
Actually, there are a whole lot of modded ships in existence, would be unsurprised if there are more than 100 (although considering how many ships exist total, that's not all that large a number). However, most mods are less extreme, and aside from their images, ships still cannot currently be modified as those methods were removed for being entirely unbalanced and arbitrary.
There are also quite a few, if not hundreds or thousands of ships that have slightly different stats from the original you can find in the Rules pages. if you called them "modded" ships you are not technically wrong.
I forget the stories for these, but from memory a good deal of Y-head Corvettes have a Docking Bay which isnt listed.
A lot of JM-5000s have only 2 passengers compared to the 5 they should normally be able to carry (worst mod ever?)
There are also Cloakshapes with extra hull strength I think?
Those mods are definitely not on par with a Hyper 14 YV-666 or hyper 15 YT-510 but technically they are "modified" when compared to what is in the rules. Depends on your point of view.
We've got, for some stupid reason, like 6 different MedNeb Stats from various bugs in the rules pages/during production. The actual number of "modded" vessels in the game, as Syn suggested is quite high, though I expect given how many of those are in inactive players hands, in the first rounds of combat we'll see a pretty rapid decrease in counts. You could argue that people aren't going to blow up a modded ship when they can steal it, but I doubt people are going to sit there and cross reference each stat point :p