Horn shape
H / h / h+
A Veilhorn can only have 1-2 horns, which must be located on their head. The exception is if the Polycerate mutation is present.
The horn shape is determined by the H/h/h+ gene, with the following possible combinations:
HH (straight)
Hh (straight)
hh (curved)
Hh+ (curved)
hh+ (coiled)
h+h+ (coiled)
Straight
HH / Hh
Straight horns should always be either completely straight, or have a very subtle curve to them, as shown on the examples below.

Smooth straight
HH / Hh
+
aa / ab / bb

Twirled straight
HH / Hh
+
Aa / Ab

Ridged straight
HH / Hh
+
Ba / Bb

Spined straight
HH / Hh
+
AA / AB / BB
Curved
hh / Hh+
Curved horns should have 1-2 curves, either forwards or backwards (or both), but may not coil around itself.

Smooth curved
hh / Hh+
+
aa / ab / bb

Twirled curved
hh / Hh+
+
Aa / Ab

Ridged curved
hh / Hh+
+
Ba / Bb

Spined curved
hh / Hh+
+
AA / AB / BB
Coiled
hh+ / h+h+
Coiled horns causes the horns to be so curved that they coil around themselves into spirals.

Smooth coiled
hh+ / h+h+
+
aa / ab / bb

Twirled coiled
hh+ / h+h+
+
Aa / Ab

Ridged coiled
hh+ / h+h+
+
Ba / Bb

Spined coiled
hh+ / h+h+
+
AA / AB / BB