can we use "can", "could", "may" & "might" to ask for suggestions"?

Question

Note: make sure you distinguish "make suggestions" & "ask for suggestions"

-make suggestions: eg: you could stay at home if it rains

-ask for suggestion: eg: "Where shall I put the vegetables?"

It seems that some resources say "we use shall or should to ask for suggestions, but not can, could, might or may"

Eg:

What shall I do with these boxes?

Where should I put the vegetables?

But, this dictionary talks about "could"

Could: used in speech to make a polite request or suggestion .... “Where can I stay?” “Well, you could always stay with me.”

They say "could to make suggestions" & they don't say "could to ask for suggestions"

We can use "could" to make suggestions, eg: you could try glue if tape doesn't work

So, in this example

A: “Where can I stay?”

B: “Well, you could always stay with me.”

could in "you could always stay with me.” make a suggestion, but see the “Where can I stay?”, it seems that can in this case is asking for a suggestion.

My question is:

can we use "can", "could", "may" & "might" to ask for suggestions"?

Answer 1

The rule children are taught is that "can" implies ability and "may" asks for permission

Q: Can I have dessert now?
A: Yes you can, but you may not.

In terms of a suggestion, it depends more on how your question is worded and also the context surrounding the question. If you are coming into the house laden with groceries, you might ask

Where shall I put the vegetables?
Where can I put the vegetables?
Where may I put the vegtables? Where might I put the vegetables? (this one is a little stylized)

All basically mean

Where do you want the vegetables to be put?

The hard and fast rules you cite, really aren't so hard and fast.

Could and should might be used to ask for suggestions

What could it possibly mean?
Should I be formal or casual?