That does not exist in 5e.
In general, if you need to touch the target, this is handled as part of the spell cast, with a melee attack to determine if you succeed. For example, shocking understanding:
Lightning strikes from your hand to strike a creature you are trying to touch. Perform a melee attack against the target. You have an advantage in the attack roll if the target wears a metal armor. In a stroke, the target receives 1d8 of lightning damage and cannot react until the beginning of its next turn.
If the spell attack fails, the spell does nothing. Note that, since you are accustomed to the mechanics of the third edition, spell attacks are launched against the normal AC of the target.
There are some exceptional cases:
Spells that act as weapons
This includes flame blade Y vampire touch. These also use melee attacks, but the spell lasts for a while and allows you to perform additional attacks during that time.
Spells that increase weapon attacks
The various paladin spells "hurt" as brand hit These are all verbal-only bonus action spells (so you can cast them with a weapon in your hand and attack immediately) with a duration of 1 minute. Once the spell ends, the next time you hit someone with a melee weapon, the spell adds damage and other effects. This is possibly the most similar to the "sustained touch spell" of the 3rd edition: you make your attack, and if you fail, it is transferred to your next attack. However, they are not "tactile" spells; You keep doing normal gun attacks.