What do you think are the chances of being bumped up on the day?
You say "bump" but it's not the same thing as being upgraded.
being upgraded means you get a better seat on the same flight
being bumped means that you do not get to board your flight and you're given a compensation (upgrade, money, free ticket) when you accept to take the next flight.
I have heard of people getting upgrades by surprise and of people asking for one and getting it.
the odds of getting an upgrade decreases as the number of passengers in your party increases.
There are very few 1st class or PE seats when compared to economy
If you're a solo traveller, there is a higher chance to be upgraded when there is an empty seat in 1st/BC
If you're travelling as a family with kids, it is less likely that there will be enough empty seats in the higher class to accomodate your whole party. So it is very less likely to be upgraded.
There is no rule for upgrades. Sometimes the rep at check in will "fall in love" with you and upgrade you or put you on the upgrade list.
Sometimes asking might get you something, but most of the times the reps will refuse upgrades to people who ask, and upgrade people who've been nice and didn't ask.
It also depends on how full your plane is. If there are seats left in economy, the odds to get an upgrade is almost 0
The main reason you could get upgraded is that the airline needs your seat for a pax who has been bumped, or who missed their flight and is rebooked on your flight. They will offer you an upgrade (if a seat is available in a higher class) in order to give this pax a seat.
Online check in is hit or miss. Sometimes airlines release seats that were not available for advance seat selection, and you may get a PE seat free (again very unlikely to get them if you're not a solo passenger, and you need to check in at the very start of the check in window), and the best chance to get upgraded are by checking in at the airport, but too many variables (you'll most likely be offered to be bumped onto another flight - possibly the next day for transatlantics) nothing is predictable
But the higher the number of passengers in your party, the lower your chances for an upgrade.