1.If you cannot win a 5 v 5 team fight. - Split Push!

This one is self explanatory. If Your team fed just one meal too many and enemies are now way too strong, a good option might be split pushing. Of course there are quite a few check marks to do check before attempting to split push. A few examples:

  1. Can the other 4 players on your team at defend on their own 4v5?
  2. Do you have enough damage to clear waves and take turrets in a reasonable amount of time?
  3. Can you see at least a few of the enemy team on the map, especially any fed enemies?

If you think it's a green light, go ahead then, push away!

2.You win any 1v1 - Split push!

Sometimes even if You could win a 5 v 5 team fight, You still want to split push sometimes. Some enemy team compositions are very nasty to take objectives from. For example if you ever tried to take a tower away from Anivia or Lux, You will know the feeling of every creep wave demolished by these champions. A good way to break a defense like that is through split push. Now You will have to check if you have teleport, to be able to join the fight at baron if it erupts suddenly. Because the enemy team won’t sit idly while you take their base. They will either try to force an objective on the map, or collapse and try to kill you instead. In those scenarios it is way better to have your teleport ready.
If there are no objectives available and You are very strong so You can beat anyone 1 v 1. Then you can force enemies to constantly send their strongest member to defend against you. That relieves pressure from your team and lets you negate any strengths the enemy team might have.

3.Your team keeps dying and you keep losing out. - Do not split push!

Sometimes every possible reason might tell you to split push, but every time you do, your team just gets into a fight, dies and you lose towers and objectives. There are a lot of things to do, split pushing might even be a good option in some of these cases, but try to group with your team. Sometimes it is a lot better to group with your team and fight together than to try to farm it out and get map pressure and win the game through that. Not every teammate will understand what you are trying to accomplish, not every team mate is aware of the potential of a well done split push. They just want to fight and kill towers with the team. In that case, it will just be wrong to try to split push and leave your team fight 4v5.