I'm going to show you a conversation I had with one of the resident goldfish with plants experts, her name is tithra
Hi there! Not all goldfish are crazy plant eaters like most people would have you believe. Single tails do seem to be a bit more interested in plants than fancies as a whole, but this is a huge generalization. There are many people I know with single tailed fish who successfully keep a planted tank.
Ultimately, it really just depends on the fish. I have had 3 large fancies so far and none of them have messed with the plants. I am able to keep a wide variety of plants successfully including many that are traditionally considered non-compatible with goldfish.
Based on my experience as a moderator at kokos goldfish forum the last couple years (go check it out if you haven't already), I think there are three basic categories when it comes to goldfish and plants: 1) those that attempt to eat every green thing in the tank including the more traditionally goldfish compatible plants like java fern and anubias, 2) those that eat certain plants but not others. This comes down to trial and error, but usually these fish go for more delicate leaved plants and will leave those with thicker, tougher leaves alone, and 3) those that could really care less about the plants.
Numbers 1 and 3 are certainly the minority, with number 2 being the majority of goldfish. But the only way to find out what type you have is through trial and error This is why I always encourage goldfish keepers to not limit themselves simply because they have goldfish. I really hate it when I see people discouraging someone from getting a particular plant simply because they have goldfish. If there is a plant that you like the look of, try it out and see. If your fish eat it, then you know to try something else next time. If they don't, then yay! lol.
There are certainly plants that are more likely to be eaten than others though. Plants that are least likely to be eaten in my experience are java fern and anubias, these tend to hold up to most any goldfish, although I have seen some that will rip them apart. After that in the small chance of being eaten category are swords, crypts, hygros, vals. In the more likely to get eaten category are finer leaved plants like wisteria, water sprite, pennywort, ludwigias, rotalas etc. And in the most likely to be eaten category are the very delicate carpeting plants like HC and DHG.
Even if plants aren't eaten, keeping stem plants rooted initially can be tricky. My fish are pretty good, but they do root around in the plants a bit looking for food, so when I am first setting up a tank I am often doing some replanting at water changes because some plants will get pulled up. So, plants that are quick to root and that have good heavy root systems can be a good choice (swords, crypts, hygros, even ludwigias can have a pretty sturdy root system that develops quickly).
Keeping your fish well fed can be another factor that contributes to plant success. There is always all this worry that we overfeed goldfish, but I actually find in many cases we underfeed. Small/young goldfish need lots of small meals a day for optimal growth, we are often taught to feed 1-2 meals, whereas 3-5 smaller meals is more ideal. With larger goldfish we often do not change the way we are feeding as they grow and we end up underfeeding because we are still giving them the same amount as we gave them when they were little. You can actually use a kitchen scale to weigh your fish to objectively measure how much food they should be getting a day (.5-1% of their body weight in pellets is typically recommended). When I weighed my fish for the first time I was really surprised to find I was grossly underfeeding. I now weigh them every 2-3 months to get an estimate on how much I should be feeding. If they are well fed, interest in plants as food tends to go down for some fish (although not all).
Hope that helps some. Check out kokos if you haven't before. Most of our members keep a planted tank of some sort. Few have elaborately planted tanks like you see here but some do, and almost everyone keeps at least some live plants instead of fake. Even people with bare bottom tanks at least keep anubias tied to rocks