I think this type should have a Semigroup instance, but I'm not sure if this is the right one. There are at least two more:

(Semigroup a, Applicative f) => Semigroup (NonEmpty f a)
Alternative f => Semigroup (NonEmpty f a)

Ok, I'm not sure the difference between those instances really. I just needed Applicative's pure to get it to work and played around until it compiled. I'll be happy to update if you tell me which is more appropriate.

Just to compare, here are all three versions (assuming this is what @paf31 meant):

instance i1 :: (Applicative f, Semigroup (f a)) => Semigroup (NonEmpty f a) where
   append (a :| fa) (b :| fb) = a :| (fa <> pure b <> fb)

instance i2 :: Alternative f => Semigroup (NonEmpty f a) where
  append (a :| fa) (b :| fb) = a :| (fa <|> pure b <|> fb)

instance i3 :: (Semigroup a, Applicative f) => Semigroup (NonEmpty f a) where
   append (a :| fa) (b :| fb) = (a <> b) :| (append <$> fa <*> fb)

The first two are the same for f = Array:

("1" :| ["2", "3"]) <> ("4" :| ["5"]) == "1" :| ["2", "3", "4", "5"]

while the latter fulfills:

("1" :| ["2", "3"]) <> ("4" :| ["5"]) == "14" :| ["25", "35"]

I think the first option (Applicative f, Semigroup (f a)) is more appropriate. It mimics the behaviour of Array and other container types where (<>) concatenates the two containers rather than the individual items within.

Also, in the last example, when the two containers don't have the same length, the behaviour is rather weird. Especially with regard to the last element in the resulting list. You're probably better off doing something based on map+zip.

Closing this until we can decide which instance is the right one.