Many people claim the Aerondight sword to be the best in the game. Problem is it has no sockets.
I assumed that to enchant a weapon, in my case to add three sockets to this sword, one had to visit the runewright but this does not seem to be the case. Some guidance would be appreciated.

Aerondight is not a normal sword, and cannot be upgraded by any smith or runewright. It already has bonus powers. From the description:

"Each blow generates charges which increase sword damage by 10%. Charges are lost over time or when receiving damage. A fully-charged sword always deals critical hit damage.

Killing a foe with a fully loaded sword will expend its charge to permanently increase the weapon's damage. Damage is increased by: 0 (to a maximum of 10 - increasing along with your character level)."

In other words, you have to use it to keep it upgraded.

This is not correct. Aerondight can have sockets added to it by the runewright in HoS. I've done this myself, in a completely unmodded game. Three sockets can be added, although doing so is not cheap.

It may be necessary to complete all of the runewright's "upgrade" quest chain first; I've never tried to enchant anything prior to doing so, however. Note that adding sockets to an item is not the same as inscribing a runeword, and is done on a separate interface tab.

You're probably right. I was assuming the OP meant regular runes, so the use of the word "runewright" was incorrect on my part.

I finally got the Runewright to add three sockets to my Aerondight.
I'm still a bit sceptical to replace my current sword with this.

Currently equipped: +330 armor piercing
+ 30% quen intensity
+75% critical hit damage bouns
+15% critical hit chance
+ 15% chance to stagger
and this does not include the three runestones attached.

The Aerondight has only +9 damage but no other features.
I understand that it sort of upgrades and levels up as you level up
but I still feel safer with what I have tried and tested.

Aerondight has the feature that damages goes up +10% to +100% depending on charges, and at 10 charges also gains a 100% critical-hit rate. Thus, while its uncharged DPS isn't amazing, after a few hits during the fight it becomes rather good.

It's good *if* you're rarely getting hit for health damage (a hit that merely breaks Quen but didn't hurt you doesn't count) because getting hit for health halves your hit counter.

All a bit technical. What increased damage? damage to armor? better sign damage? Chance to freeze etc?

So based on the specs of my current sword would you personally swop it for the Aerondight?

I think it depend on your playstyle, I choose to have a set(griffin) for the bonus of it...

Aerondight is the best pure damage sword in the game. It benefits from high base damage, can do increasing damage when used in battle due to the previously mentioned "charge up" mechanic, and can have its base damage permanently increased by killing an enemy with a fully charged blade. Because of this, in can dynamically scale with your level provided you make the time investment to use it.

However, it does not confer any other benefits that some blades with static attributes possess. Damage is all it does. It has no extra critical effects like stun, burning, poison, etc. It has no armor piercing or experience boosting qualities. It doesn't contribute to set bonuses like the witcher gear swords do. And the sword's charges decay over time or when taking damage in combat, so it can be tricky to upgrade.

The question of "which is best" depends on how you like to play the game. Ultimately the purpose of a sword is to kill things, and that means doing damage. However, special effects like freeze, stagger, and so forth can help tremendously when they work. Aerondight requires heavy use to maximize its benefits. Other weapons are as good as they can be when you get them. So your decision comes down to choosing a more versatile blade with an immediate payoff or a more damaging one that needs more development.