RAW : It's complicated
The rules for cantrips state, emphasis mine :
A cantrip is a spell that can be cast at will, without using a spell slot and without being prepared in advance. Repeated practice has fixed the spell in the caster's mind and infused the caster with the magic needed to produce the effect over and over. A cantrip's spell level is 0.
The argument over whether the emphasised part is a list, whether the "can" implies a possibility only for the first element or for all of them, and all those debates will lead us nowhere, simply because the english language here is ambiguous. So let us focus on other, less ambiguous rule texts that may point us in the right direction.
Take a look at this extract from the rules on Spellcasting for Wizard :
PREPARING AND CASTING SPELLS : The Wizard table shows how many spell slots you have to cast your spells of 1st level and higher.
As stated in the description at the start of this answer, cantrips are level 0 spells. Because of those rules, you can neither "prepare" cantrips (you don't have a cantrip prepared in the first place, you "know" it), nor use your spell slots to cast them.
Moreover, there is also this rule text a bit further :
Vou prepare the list of wizard spells that are available for you to cast. To do so, choose a number of wizard spells from your spellbook [...] The spells must be of a level for which you have spell slots.
Since cantrips are not in your spellbook, you definitely cannot "prepare" them.
This kind of text is found in the spellcasting rules for other classes capable of casting spells. Here are the relevant extracts :
SPELL SLOTS : The Bard table shows how many spell slots you have to cast your spells of 1st level and higher.
PREPARING AND CASTING SPELLS : The Cleric table shows how many spell slots you have to cast your spells of 1st level and higher.
PREPARING AND CASTING SPELLS : The Druid table shows how many spell slots you have to cast your spells of 1st level and higher.
Fighter (Eldritch Knight Archetype) :
Spell Slots. The Eldritch Knight Spellcasting table shows how many spell slots you have to cast your spells of 1st level and higher.
Oddly enough, Paladin has a slightly different rule text :
PREPARING AND CASTING SPELLS : The Paladin table shows how many spell slots you have to cast your spells.
However, this does not matter since Paladin does not have cantrips in the first place.
SPELL SLOTS : The Ranger table shows how many spell slots you have to cast your spells of 1st level and higher.
Rogue (Arcane Trickster Archetype) :
Spell Slots. The Arcane Trickster Spellcasting table shows how many spell slots you have to cast your spells of 1st level and higher.
SPELL SLOTS : The Sorcerer table shows how many spell slots you have to cast your spells of 1st level and higher.
... except Warlock is in an odd spot.
Just like Paladin, Warlock has a slightly different text :
SPELL SLOTS : The Warlock table shows how many spell slots you have.
I haven't found anything else in the rule text to either confirm or deny that Warlock cantrips may or may not be used with a spell slot. It seems like for Warlock, we're back at the "what does can mean in this context?" debate, which means that RAW is still unclear for this class specifically.
A possible counter argument
As highlighted by multiple people in comments and chats discussing this topic, the rules on upcasting spells states the following :
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. [...] Effectively, the spell expands to fill the slot it is put into.
If a cantrip was successfully cast using a spell slot, because of this rule, its level would become the one of the spell slot used, and it would be possible to use it with a spell slot.
However, for the spell to assume this higher level, it needs to be cast successfully in the spell slot. The rule mentioned for each class only allows you to use spell slots to cast spells of 1st level and higher, which cantrips, being level 0, do not qualify.
For a spell to be upcast and assume a higher level, it needs to be cast in a spell slot. But to cast a spell in a spell slot, you need it to be 1st level or higher. In the case of a level 0 spell, the snake bites its tail, and it is effectively impossible to upcast a cantrip.
The issue of "When"
It has been brought to attention that this last text also has an issue with the ambiguity of english language.
This phrase is formatted as "When X happens, Y happens". I naturally interpreted that as cause-consequence : if X happens, then it causes Y to happen. That is why I concluded that if X cannot happen, then Y cannot happen either. However, it would seem that this formatting could also be interpreted as synchronous : if X happens, then Y also happens at the same time.
In that second interpretation, instead of a cause-consequence reading, we have two independant actions happening at the same time. This means that the cantrip could take on the level of the spell slot the instant you cast it, and thus become a "1st level or higher spell". This would then fit the prerequisite for using a spell slot I quoted for most classes.
Both interpretations of this rule text seem valid. In the end, we're back to ambiguous english language.
Both the base rules for cantrips and the specific rules for casting spells for most classes are ambiguous as to whether you can or cannot use a spell slot to cast a cantrip, because the language used can be rightfully read in opposing ways.
What about Rules as Intended?
It seems quite obvious that the intent was that cantrips cannot be cast using spell slots, from the idea itself never being mentioned in the rules, or that all cantrips lack any upcast effect, or even the fact that the situations where this is beneficial are very rare, if they even exist. The tweet quoted in the currently accepted answer, even if unofficial, points in that same direction.
So, should I allow it?
As a DM, I see no good reason to not allow it, although the cantrip itself wouldn't be changed. If the player wants to trigger an effect that depends on the use of a spell slot, they should always be better off using an actual spell rather than a cantrip. But since there's nothing gamebreaking about this, I'd let them go for it.