Surtr Solo (S10)

Surtr is a very powerful solo laner, but lacks sustainability other than the meager 1% health tick from his passive. This is why I choose to always start with Soul Eater, making him an incredibly difficult target to take down. He can easily take on Bellona/Cu in a fight when Soul Eater is online, even going positive in health in the heat of battle while fighting under a minion wave.

Starting at lvl 1, go for your 1 ability. The 2 is the only other one that competes in dmg when it comes to lane clear, but you want to get your 1 going for your imp and fire DOT, helping with the early JG clear with your mid and JG. The 2 is not good for a start, as the only thing you can pick up is other minions, making your early JG clear slow. Take it as your lvl 2 upgrade. From the start, you should go for Soul Eater ASAP. Getting that online will make you survive solo and even ganks. As soon as that item is finished, I personally go straight for Persistent Teleport, making my lane backing more often and more reliable. Then going for the Core build, with situational differences in the build.

UPDATE:
After further play, testing and consideration, I have found that my start wasn't as strong as it could have been in terms of abilities. I made a quick judgement on the 3 ability, but it is clear to me now that starting with your 3 is clearly the best option at lvl 1. with updates to the JG starts and depending on your compositions, getting down the JG camps with blue, yellow and red is paramount to all sides of your team doing well. Doing the max amount of AOE dmg, no matter how small in that JG camp fight is a necessary evil. When you get to lane on first wave is when it gets dangerous to use your 3. I personally do not like positioning myself in the center of the enemy minion wave, potentially hitting the enemy god with my 3, then taking the brunt of aaaall the minion dmg and the god himself (if they are smart to realize the trouble you are in). But, after hitting lvl 2 with your JG camps dead, I get my 2 next, to fling a melee minion in lane to the archers for some more AOE lane dmg, while carefully positioning and judging when I can clean the rest up with my 3. After hitting lvl 3, follow the Surtr Combo section of this guide for the typical fighting style.

Here is how you should play Surtr:

Lane Clear - Start by using 1 and hitting either the god in lane or the minion with the lowest health, summoning an imp. Use 2 to grab the imp that is in range, and throw from either side of the minions to hit all the rest of the minions, and letting the imp impact area hit and explode on the majority of the minions (preferably archers). Use your 3 while in the center of the pack of minions, hitting all minions with the outer ring to kill them all down the rest of the way.

Boxing in Solo - While keeping lane clear in mind, you should attempt to hit the enemy god with your imp as much as possible. It deals a lot of dmg and stuns the target, setting up for you to hit them with another of your 1s. Keep them in line to either force them to box you, or keep them at bay with your 3. Use your ult to either gank mid or assist a fight in the JG. It has a long range and does a massive amount of dmg. And, depending on your current health, it is a great way to dive towers and get a kill secure.