If you level up a Sorcerer new spell levels do not appear.  (It knows that I know more spells, but doesn't add the new level I can learn from)
If you go to View character sheet, then back into Edit the new spell levels will load.
My Sorcerer is a Divine Origin subclass, in case that has anything to do with it.

If a user didn't notice that they should have access to a new spell level they would be inclined to take a new spell at a lower level giving themselves a handicap.

That first time you hit Add Spells it loads (like an ajax call), but it doesn't reload when you level up, and there's no manual button to force it to.

Changing to another section (like Description or Equipment) then back doesn't make the spell list reload (by design I assume, and I'm OK with that). That's why you need to go out to View character then back into Edit, that triggers the spell list to reload when you go there.

Mine levels up perfectly fine.

Sorcerers gain access to a new spell once every level until level 10 then it is once every other level until they reach 15 spells total (not including any extra spells granted by other class features).

You gain access to new spell levels every 2 levels until you reach level 9 spells (so you can cast level 2 spells starting at class level 3 for example).

When you level up you can change one spell you know for another spell of any level you have access to.

To level up you need to access the Character Builder by either clicking the character name on the sheet to open a menu and choosing "manage level" or by clicking the symbol on the top-right of the sheet that looks like a hammer and anvil. This takes you to the Character Builder. On the builder go to the Class page and use the drop-down on the top-right to increase the level. Any new features will appear on the class info tab and you can click the spells tab to add or change spells. Click the sheet icon when done to go back to your character sheet.

Let me walk you through my actions:

Create Character
Select any race
Select Sorcerer
Select Storm Sorcery subclass (or whatever)
Go to Abilities
Standard Array
    Str 10, Dex 14, Con 13, Int 8, Wis 12, Cha 15
Go to Class
Go to Spells
    Add Spells (any 4 cantrips, any 2 1st level spell)
Bump your level to 2
Known Spells will bump to 3 (which is correct)
Learn any 1st level spell
Bump you level to 3
Known Spells will bump to 3 (which is correct)
But the 2nd level spell filter does not show up, neither do 2nd level spells

I'm on Firefox on Linux (Lubuntu)

I'll try it on Chrome to see if that's a factor.  It does the same thing.

You can jump to level 11 and you'll still only see cantrips and 1st level spells.  Going to description and back doesn't help.  You need to leave the character builder (ie, go to View character sheet).  Then when you go back to Add spells it will reload correctly.

I play sorc chars and have used D&D Beyond in Firefox, IE and Chrome and not encountered any problems as you describe. So I tested on my android phone using the standard browser and firefox and again no issues at all.

Might be your browser settings. Are you using anything homebrew? If so, remove it and see if there's a difference. Try making a different sorc using only standard official things, see if you still get the issue.

And you can consider adding a link to the character sheet here so mods can take a look at it.

I know that addons can cause issues.  But my addons are different between browsers.

An F5 (refresh) does work.

ddb.ac/characters/8522045/7BDk7s

i don't know that anything will be visible there though.  It's just that when i change the level it doesn't refresh the page.  I'll mess with my addons and bring up F12 to see if something it dying behind the scenes.

When you click manage levels or click edit character, add the levels on the Class Features tab. Then move to the Spells. If you add the level on the Spells tab you need to refresh. If the F5 solves it, just use that and problem solved.

You're not alone in this man, I have to refresh every time I level to make sure I have the correct spell levels available. It's pretty annoying. 

Mine knows that I can cast third level, but won’t let me pick. Also annoying for sorcerer, you can add spells acquired through other means. There needs to be “add manually”  

Quote from ElricTruridge >>

Mine knows that I can cast third level, but won’t let me pick. Also annoying for sorcerer, you can add spells acquired through other means. There needs to be “add manually”  

So to confirm you are now 5th level in Sorcerer but it won't let you pick 3rd level spells and have 5 or less known spells (as, at 5th level, your max known spells is 6)?

Also: Sorcerers are not wizards. They add spells through level up and cannot just "gain" them through other means with very specific exceptions such as subclass features or feats. The tools here are focused on the official rules, so that's why there is no ad-hoc adding of known spells.