after which one of the verbs you use an ing verb form?

Question

UPDATED: I do really appreciate your explanations and taking time. Nevertheless, none of them isn't my answer. I, again, try to explain my specific question:

As I have already surfed the Internet and some dictionaries, I have realized the fact that, for instance the bold part, either ing form of a verb or an infinitive with to could appear after cannot. So, I am wondering the other same rules for italic parts.

cannot tolerate

cannot resist

cannot help

cannot bear and cannot stand both of them need either ing form or infinitive with to

cannot avoid

Any help would be greatly appreciated

Answer 1

The verb which follows a modal verb (can/could, may/might, must, shall/should, will/would) in a verb group is always a ‘bare’ infinitive—that is, an infinitive which is not ‘marked’ with to. (The presence of not or an adverb does not affect this rule.) Marked infinitives and partiples cannot occur in this position.

^okShe cannot tolerate^{bare infinitive} his rudeness.
^∗ She cannot to tolerate^{marked infinitive} his rudeness.
^∗ She cannot tolerating^{present participle} his rudeness.
^∗ She cannot tolerated^{past participle} his rudeness.
^∗ _{marks a form as ungrammatical}

However, modal verbs can combine with progressive, perfect and passive constructions. When that is the case, it is only the first auxiliary after the modal which is cast in the infinitive. Subsequent auxiliaries and the main or ‘lexical’ verb take the form required by the construction. (But none of these constructions takes a marked infinitive).

PROGRESSIVE: ^okShe cannot be^{bare infinitive} tolerating^{present participle} his rudeness.
PERFECT: ^okShe cannot have^{bare infinitive} tolerated^{past participle} his rudeness.
PASSIVE: ^okHis rudeness cannot be^{bare infinitive} tolerated^{past participle} by her.
PERFECT PASSIVE: ^okHis rudeness cannot have^{bare infinitive} been^{past participle} tolerated^{past participle} by her.

Answer 2

First off, it is absolutely irrelevant if you use "can" or "cannot" before the verb, which–as StoneyB correctly pointed out–is always the bare infinitive when you use a modal verb.

However, if I understand your question correctly, you ask about how to know if you have to use the gerund or infinitive after the main verb.

I'm afraid but you have to learn them by heart. There's not even a rule of thumb. To make it worse, there are four types:

• verbs that take gerund only
• verbs that take infinitive only
• verbs that take both, with similar or identical meaning
• verbs that take both, but with different meaning

Example:
"I stopped smoking" means that you don't smoke any more; "I stopped to smoke", however, only suggests that you halted something, which is not mentioned, for a moment in order to smoke.

It's pretty impossible to go into details about each and every word but when search the Internet for "verb + infinitive + gerund", you often find the answer, especially for those verbs where gerund and infinitive uses have different meanings.
Additionally, you often find examples in learner's dictionaries.

Answer 3

The verbs
tolerate
risk
help
avoid
are all followed by the -ing form.