My most used weapon at the moment is Miraak's sword, because it swings very fast compared with other swords. This means I can very quickly apply two poisons to an enemy, leaving him paralyzed and taking 50 damage each second, then move on to the next one.
For my Warrior, I use a custom made Ebony Mace with Shock Damage. I love the look of the Ebony Mace and it has one of the best stagger rates in the game, making it great for a battlemage type character.
My most used weapon would be the Nord Hero Sword with a burn/absorb health enchantment. I particularly love the ancient weapon look, but it also wins a lot of points for me by having no sheath.
If asking about favorite weapon - Dragonbone greatsword. That thing just looks badass.
I tend to restart A LOT and still haven't beaten any of the DLCs (or even started Dragonborn). However, the one staple of almost every character I make is joining the Companions and in the process using the Skyforge steel sword. It's strong at low levels and even acceptable with high smithing at the high levels. It also looks especially cool with the aMiridianborn textures from the Nexus.
Ebony weapons tend to be my most used weapon in the game. However my most favoured weapons are the glass variant. My current character: a Breton knight of particular Altmer descent uses a custom sword called Valiant-enchantment free (for the intended use of elemental fury).
Dwarvern Enhanced Crossbow enchanted to be the bane of my opponent's existance, sure it makes noise so sniping without getting caught in a long fight would be tough, but that is not my playstyle. My second favorite is Ahzidal's Ring of Arcana, because sometimes the world just needs to burn, and it is the perfect tool for any pyromaniac.