Can you float mana from one phase and spend it in the next phase?

Mana stays in your mana pool until something causes it to empty. Under the current rules (since Magic 2010), mana no longer automatically empties at the end of every phase or step — it only empties when the stack is empty and all players pass priority, causing a spell or ability to resolve or a phase/step to end (CR 500.4, CR 704.5d... specifically CR 500.4).

In practice, mana empties at the end of each step or phase when the game checks if the active player can proceed. So if you tap lands for mana during your main phase and then your main phase ends, that mana is gone before your combat phase begins (CR 500.4). You cannot simply 'save' mana from one phase to spend freely in the next.

However, there are situations where mana can carry over. If you generate mana during a phase transition — for example, with a triggered or activated ability on the stack that crosses a phase boundary — mana produced before that ability resolves can persist. More practically, some effects explicitly let mana persist (e.g., cards that say 'this mana doesn't empty').

Concrete example: During your precombat main phase, you tap a Forest for one green mana but don't spend it. When your main phase ends and combat begins, that green mana empties from your pool before you receive priority in the beginning of combat step. You cannot use it to pay for a combat trick.

So floating mana across phase boundaries is not possible under normal circumstances; the mana pool empties as each step or phase ends (CR 500.4).