Dang ok I have been meaning to make this post since forever, actually. Here we go.
I do personally own a ram skull, and have studied it a bunch especially with reference to drawing my curly-horned trollsona, so that helps. One thing you can do is roll and twist some paper, which is what I often do for Eridan’s horns, because wtf Eridan’s horns.
I like to draw all the horns as variations on a spiral, even Karkat’s, so when I’m not being lazy that’s how I draw the grooves. But not all horns are like that, so, do what you want.
Trigger warning: searching for nondomesticated ungulates is pretty likely to show you hunting pictures with dead animals.
For reference pictures, google search curly sheep horns. Note that sheep horns grow out in a coil, like a spring, and taper most sharply at about the last quarter of the last curl. Sheep horns have a really intense bumpy fingernail sort of look, with grooves and bumps wrapping around the circumference of the horn rather than spiraling along it.
Cow horns taper a lot more— to reference Tavros’s horns, I like to google highland cattle, since they have nice wide racks that curve out, up, and forwards. Longhorns also have, obvs, nice long horns. When Tavros’s horns are drawn really flat along a z-axis it just looks fucking strange to me. They also often have streaks and stress lines down from the tip, back along the horn, though they also have a bit of the lumpy fingernail effect of sheephorns at the base. They do spiral, but a lot more subtly than a sheep’s.
The trolls with simple horns (most of them tbh) can be referenced off goats—there’s a lot of debate over where exactly Sollux’s are in relation to each other. Are they all set along the same arc? are the smaller set inside and forward? outside? back behind??? who knows.
to actually get pictures of actual goats with actual horns, search ‘mountain goats fighting’. also, the big brown guys with the heavy corkscrew horns are not goats, they are bighorn sheep. google is generally totally shit at telling sheep apart from goats.
I think Gamzee’s horns are shaped very much like a lyre (an instrument, not an animal (unless we’re talking about lyrebirds (we’re not))), but judging by how GHB’s got some extra wibblies it’s likely they’re like the kudu's. You can also look up markhor goats, though their horns are very flat and ribbonlike, and troll horns seem to be mostly round in cross-section.
Eridan’s horns could be along the lines of this same corkscrew growth pattern: one twist to account for each change of direction, though Dualscar didn’t have any more bends, so maybe they’re more like a bongo’s. Who knows.
kudu with short horns^
bongo with regular horns^
finally, this is a capra ibex:
and ibex are not to be confused with oryx:
scimitar horned oryx^
who are frequently confused with gemsbok.