First draft at some abilities for the Aasimar, 1 of them is based off of the DMG, 3 are based off of the subraces from Volo's Guide to Monsters, 1 is based off of the 4e Deva.
Aasimar Characters
Ability scores & classes: Aasimar often are called upon to be champions, so many display a natural aptitude in athleticism or leadership. So most Aasimar put their ability bonuses in Charisma, Wisdom or Strength.
As a result many Aasimar adopt the path of the Paladin, where they can bring their might and divine light into a cause. Since their celestial nature leads them to be more in touch with the divine, Aasimar also commonly become Clerics. For other Aasimar they draw upon the spark of inspiration to become Bards. As many celestial beings such as Guardinals and Celestial Eladrin are idealized spirits of nature, Aasimar often become Druids too.
Age
Most Aasimar age at the same rate as Humans or their mortal ancestry does. Some Aasimar might be long-lived or ageless.
Size
Most Aasimar are medium size. Some Aasimar are small size.
Speed
Aasimar have a walking speed of 30 feet
Type
Aasimar are Humanoid
Languages
Aasimar understand Common and one other language, often one of their mortal ancestries or Celestial.
Darkvision
Aasimar have Darkvision of 60 feet
Celestial Legacy (Legacy trait)
Much like Tieflings there are varieties of the powers an Aasimar can be imbued with. The nature of the celestial can shape the temperament of an Aasimar, revealing a little of the soul when such powers are used. Aasimar often radiate light and heat or faint music might be heard when using what Aasimar often call Revealing the Soul or Calling the Spirit. For some Aasimar it's a struggle between their mortal selves and their celestial selves when using such powers. An Aasimar picks from one of these Legacy options below, Charisma or Wisdom is the ability score for spells and saving throws DCs with these powers.
Optional Rule: Custom Legacies - For custom legacies (with the DM's approval) they generally follow this format, one Cantrip at the start, at 3rd level they know one 1st level spell that's enhanced to a 2nd level spell slot if possible and can be cast once before requiring a long rest, and at 5th level they gain knowledge of a 2nd level spell that can be cast once before requiring a long rest to cast again. The spells should be from either the Cleric or Paladin spell list.
Legacy of Light - The spirit of the Aasimar glows with the light of the divine serving as a beacon of hope and expectations the celestial inside aspires to. The powers of this legacy are: the cantrip Light, at 3rd level Lesser Restoration once before requiring a long rest, and at 5th level Daylight cast once before requiring a long rest.
Alternate Spells: Replace Light with Guidance and/or replace Daylight with Beacon of Hope.
Legacy of the Protector - The celestial inside seeks to defend those around it, driving the Aasimar to become a champion for a cause. The powers of this legacy are: the cantrip Resistance.
Healing Hands: with an action you can touch a creature a heal a number of hit points equal to your level, you can't use it again until you complete a long rest.
Radiant Soul: At 3rd level you can transform yourself into an angelic being as you radiant with light and glowing wings sprout from your back. For 1 minute or until you end it as a bonus action, you gain a fly speed of 30 feet, and once per round to one target you can deal extra radiant damage equal to your level with an attack or spell. Once you use this you can't use it again until you complete a long rest.
Alternate Abilities: Replace Healing Hands with Cure Wounds (cast as 2nd level spell)
Legacy of the Scourge - The celestial inside seeks to punish those it feels are wicked, it's a wrathful spirit that drives the Aasimar towards battle and confrontation. The powers of this legacy are: the cantrip Sacred Flame
Healing Hands: with an action you can touch a creature a heal a number of hit points equal to your level, you can't use it again until you complete a long rest.
Radiant Comsumption: At 3rd level you cause yourself to glow with blinding light glowing with bright light in a 10 foot radius and dim light for an additional 10 feet. For 1 minute or until you end it as a bonus action, at the end of your turn you and each creature within 10 feet take damage equal to half your level (rounded up) , and once per round can deal extra radiant damage equal to your level with an attack or spell. Once you use this you can't use it again until you complete a long rest.
Legacy of the Fallen - Sometimes it's a fallen celestial that is the influence of an Aasimar seeking instead to destroy the mortal world. Other times it's the nature of the mortal soul that has corrupted the celestial inside. The powers of this legacy are: the cantrip Thaumaturgy
Draining Touch: with a melee spell attack you can inflict necrotic damage equal to your level, and healing yourself with an amount of damage equal to your level. You can't use this ability again until you complete a long rest.
Necrotic Shroud: At 3rd level you cause yourself to glow with dark energy and skeletal flightless wing sprout from your back. For 1 minute or until you end it as a bonus action, once per round can deal extra necrotic damage equal to your level with an attack or spell. When you first transform creatures within 10 feet of you must make a charisma save or be frightened of you until the end of your next turn. Once you use this you can't use it again until you complete a long rest.
Legacy of the Veil - Many celestials prefer to hide their presence from mortals guiding them towards the path of benevolence. The celestial inside prefers a more subtle approach. The powers of this legacy are: the cantrip Guidance, at 3rd level Disguise Self once before requiring a long rest, and at 5th level Blur cast once before requiring a long rest.
Alternate Spells: Replace Guidance with Minor Illusion and/or Blur with Enthrall.
Legacy of Enlightenment - The celestial inside seeks to learn from and understand the world around it. Knowledge it feels is the key to enlightenment which it feels it should guide all beings towards. The powers of this legacy are: the cantrip Thaumaturgy, at 3rd level Bless once before requiring a long rest.
Memories of a Thousand Lifetimes: With a halo of light and a piercing stare you can call upon your celestial knowledge, gaining +1d6 to the roll of a skill or ability check. Once you use this you can't use it against until you complete a short rest.
Celestial Resistance (Resistance trait)
An Aasimar's nature often makes them resistant to the affect of certain maladies and elements. Pick one from among the following damage types: necrotic and radiant (both of them), or fire or lightning damage.
Alternative Abilities:
Some of these abilities might be restricted at character creation, by the DM's discretion. They might be obtainable later with the Planar Legacy feat. These abilities below can be picked instead of any of the Celestial Legacy or Celestial Resistance options.
Celestial Wings - You have a working pair of wings which may be feathered or made of energy and light. You gain a fly speed of 30 feet. This ability replaces Celestial Legacy.
Alternate rule for Celestial Wings - An Aasimar with wings gains the ability to glide, subtract 100 feet from a fall when determining damage taken, and they can move 2 feet horizontally for every foot they fallen. At 5th level they can fly with speed of 30 feet.
Celestial Alacrity - Your have supernaturally fast reflexes always rushing the aid defense of others, your walking speed is 40 feet, and when you take the Help action you may do so as a bonus action. This ability replaces Celestial Legacy.
Radiant Touch - Your hands glow with light, your unarmed strikes do 1d6 radiant damage instead of bludgeoning damage. This ability replaces Celestial Legacy.
Divine Resilence - When you are conscious, can take actions and have less than half of your hit point total, at the end of your turns you regain half of your proficiency bonus (rounded down) until you have half of your hit points. This ability replaces Celestial Resistance.