Mobile says that when you make a melee attack against a creature you don't provoke OAs from that creature for the rest of the turn. Booming Blade specifies that you make a melee attack with a weapon as part of casting the spell. RAW: Can you booming blade and avoid opportunity attacks if you have the Mobile feat?

I'd say yes.

 Yea I can't see why not, seems like a fun combo that won't break the game.

In a similar vein, would Booming blade proc Sneak Attack and or Smite?

Quote from Nike_Leon >>

In a similar vein, would Booming blade proc Sneak Attack and or Smite?

Yes - because Sneak Attack only requires you hit with an attack (and all the other requirements) and Smite says when you hit with a melee attack.

Things it generally won't work with are ones that require the Attack action - because to cast Booming Blade is a [Tooltip Not Found] action (and exception would be Bladesinger's Extra Attack).

Quote from Emmber >>

Quote from Nike_Leon >>

In a similar vein, would Booming blade proc Sneak Attack and or Smite?

Yes - because Sneak Attack only requires you hit with an attack (and all the other requirements) and Smite says when you hit with a melee attack.

Things it generally won't work with are ones that require the Attack action - because to cast Booming Blade is a Cast a Spell action (and exception would be Bladesinger's Extra Attack).

 How is the Bladesinger's Extra Attack an exception?

Because they can cast a Cantrip like Booming Blade during the Attack action by replacing one of their attacks with it. This means they satisfy having taken the Attack action while also having cast the spell.

Quote from Emmber >>

Because they can cast a Cantrip like Booming Blade during the Attack action by replacing one of their attacks with it. This means they satisfy having taken the Attack action while also having cast the spell.

 I don't see that anywhere in the Bladesinger rules. I know Eldritch Knights get to attack as a bonus action after casting a cantrip. 

Edit: I see now that it is included in the TCoE version of Bladesinger's. The dndbeyond app hasn't updated a lot of the class features with those options. Hence my confusion. 

https://www.dndbeyond.com/classes/wizard#Bladesinging

under "Extra Attack"

You can attack twice, instead of once, whenever you take the Attack action on your turn. Moreover, you can cast one of your cantrips in place of one of those attacks.

What's interesting is, with the Bladesinger you are casting a spell without taking the [Tooltip Not Found] action, but rather with the Attack action. I don't believe that there are any class or other features out there that trigger on "when you take the [Tooltip Not Found] action..." rather than "when you cast a spell...".... but if one ever comes out, keep an eye peeled 

Quote from Chicken_Champ >>

What's interesting is, with the Bladesinger you are casting a spell without taking the Cast a Spell action, but rather with the Attack action. I don't believe that there are any class or other features out there that trigger on "when you take the Cast a Spell action..." rather than "when you cast a spell...".... but if one ever comes out, keep an eye peeled 

 I think the Eldritch Knights War Magic might be such a feature. 

Phb. 75

WARMAGIC

Beginningat 7th level when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.

There are plenty of features that trigger “when you cast a spell/cantrip” or “use your action to cast a spell/cantrip.” What I’m saying is, in the same way there are some features that say “when you take the Attack action to make a melee attack...” rather than “when you make a melee attack...”... we don’t yet have an example of any features that say “When you take the Cast a Spell action to cast a spell...” rather than “when you cast a spell...” or “when you use your action to cast a spell...”

Its a minor nuance. Just interesting. 

I think it's because you can take the [Tooltip Not Found] action as an Action a Bonus Action or even a Reaction. It's too fluid to really base a feature off. I guess?

That would be my guess too. You can [Tooltip Not Found] two or three times per turn, do it in your own turn or others turns, do it as an action, bonus, or reaction... a lot of different contexts to keep track of, far more than “when you Attack”  or “when you Dodge” present. 

It is easily possible to Cast a spell four times during a single round (reaction before your turn, your turn, bonus action on your turn, reaction after your turn). 

If you multi-class, it is possible to do even more, via Action Surge or Thieves' Reflexes.

Except for that you can only use one levelled spell per round of initiative plus cantrips

Quote from JooBear >>

Except for that you can only use one levelled spell per round of initiative plus cantrips

I don't know where people get this idea but it is wrong. You can cast as many levelled spells as you have actions per round - so long as you don't cast a bonus action spell. The only rule in the books that prevents casting multiple levelled spells in a turn is the bonus action spellcasting rule. If you don't use your bonus action to cast a spell - you're perfectly fine using two Fireballs and a Counterspell in the same round if you use an Action Surge - for example.

Quote from JooBear >>

Except for that you can only use one levelled spell per round of initiative plus cantrips

 I have a graphic that shows exactly how this rule works.

Action surge, quickened and war caster are the exceptions and not the rule - Gods can cast two levelled spells per turn, mortals cannot

Say what now?