As a rogue, I've gained enough XP to level up to level 3. However, I'm not yet ready to pick what archetype would fit best for me.
Despite having gained enough XP, can I wait until later to level up my character?
I'm looking for rules to back me up, especially if it explicitly states that I don't need to level up immediately upon acquiring the XP threshold.
Best Answer
Rules-as-written, you level up right away.
The PHB states:
Note that it says "a character [...] advances in capability", not "a character can advance in capability". Contrasted with many other features that are optional, this means (RAW) that you can't delay your level-up.
Furthermore, Xanathar's Guide to Everything has a variant rule for shared campaigns, so-called checkpoint leveling (see XGtE p. 173). It's basically another alternative to XP, much like milestone leveling, so I won't go into details; its description says the following, however (emphasis mine):
This strongly suggests that leveling up as soon as you can is not only RAW, but also RAI (rules as intended).
I haven't found anything else targeting your issue in either the PHB, DMG or XGtE.
"Rules-as-fun", you probably won't run into any balancing issues as long as you talk with your DM and don't decide to level up mid-session.
Since the rest of your party probably doesn't want to wait, you'll be one level behind them. That's not a problem, it's quite common if the DM awards XP individually to everyone. However, you have to make sure to ask your DM if delaying your level-up is ok: it will affect their encounter balancing, especially if your DM normally makes sure that everyone levels up at the same time, for example by using milestone leveling.
For the same reason, mid-session-level-ups are also a no-go.
The DM may also have something planned that strongly requires you to level up. For instance, if you're level 4 and the big boss battle is coming up, you shouldn't be delaying your level-up, as level 4 to 5 is a pretty significant jump in power.
Therefore, if your DM says "no, you can't delay your level up", you should accept it without objecting—they likely have their reasons, not all of which they may be able to reveal to you.