Debug mode

Many versions in the Connexion / Connection and Plus series can be put in debug mode by shorting a jumper in their circuit. The simplest method is to use the graphite from a pencil, but this won't work every time…
You have to short a different jumper on each version, I'll write some examples here (with the side of the board you can find the jumper on between brackets):

Hyperspeed

You can make time pass very fast in your Tamagotchi if you activate this mode.
This function really helped me figure out how to get each character on several versions.

Press the Ⓒ button to switch between the normal speed and hyperspeed.
On the V4 and the V4.5, the reaction that happens when you press this button makes it impossible to switch out of hyperspeed, rendering this mode unusable. The Keitai and Jinsei series have a Ⓒ-click reaction too, but they do not have this problem: in debug mode, these reactions are disabled to allow you to switch between the two speeds.

Character choice

You can choose the character you start with in this mode. Press Ⓐ right after your Tamagotchi hatches out of their egg to switch the character you have until you find the one you want. This is the only time you can do this; you won't be able to change your character later.

You can raise character you cannot normally get during normal gameplay this way.
The Keitai series changed this a bit. You can no longer choose the Deka characters, even if they exist in the code. You must modify the save data to see them.

Care mistake counter

This counter has only been found on the Connexion V2.
You must short the J8 jumper on the screen side of the circuit to see it.
The mental ("heart"?) care mistakes are counted by "H", and the physical ("body"?) ones are counted by "B".