No they cannot. They can ask you to let them in, but you may refuse if you want.

Only if you give them permission.

From what I understand, local police need a warrant or probably cause but federal officers can do anything they please. Under the Patriot Act, they can search your home without your knowledge.

only with permission..

Laws vary world wide ... in some contries the police can break in without notice or permission, in other countries they need a warrant or permission ... so it depends on where in the world you live.

In the United States, you do not need a warrant if you have probable cause with exigent circumstances. An exigent circumstance, in the American law of criminal procedure, allows law enforcement to enter a structure without a warrant, or if they have a "knock and announce" warrant, without knocking and waiting for refusal under certain circumstances. It must be a situation where people are in imminent danger, evidence faces imminent destruction or a suspect will escape. Generally, an emergency, a pressing necessity, or a set of circumstances requiring immediate attention or swift action. In the criminal procedure context, exigent circumstances means: An emergency situation requiring swift action to prevent imminent danger to life or serious damage to property, or to forestall the imminent escape of a suspect, or destruction of evidence. There is no ready litmus test for determining whether such circumstances exist, and in each case the extraordinary situation must be measured by the facts known by officials. n each of the situations below, a police officer does not need a search warrant to conduct a search. 1. If an individual voluntarily consents (agrees to) a search, no warrant is needed. The key question in this kind of search is what counts as a voluntary agreement? In order for a consent search to be legal, the individual must be in control of the area to be searched and cannot have been pressured or tricked into agreeing to the search. 2. A police officer that spots something in plain view does not need a search warrant to seize the object. In order for a plain view search to be legal, the officer must be in a place he has the right to be in and the object he seizes must be plainly visible in this location. 3. If a suspect has been legally arrested, the police may search the defendant and the area within the defendant's immediate control. In a search incident to arrest no warrant is necessary as long as a spatial relationship exists between the defendant and the object. 4. Following an arrest, the police may make a protective sweep search if they reasonably believe that a dangerous accomplice may be hiding in an area near where the defendant was arrested. To do so, police are allowed to walk through a residence and complete a "cursory visual inspection" without a warrant. If evidence of or related to a criminal activity is in plain view during the search, the evidence may be legally seized. 5. If the police stop a car based on probable cause, they can search for objects related to the reason for the stop without obtaining a warrant. During a car search, the police are also allowed to frisk the subject for weapons, even without a warrant if they have reasonable suspicion that the suspects may be involved in illegal activities. http://www.landmarkcases.org/mapp/when.html http://en.wikipedia.org/wiki/Exigent_circumstance

In the event that you actually invite them in they may take advantage of the situation and try to search your house. Usually the police will try to trick you by trying to talk to you inside your home. One officer will ask you questions to distract you while their partner sneaks about. This will give them a chance to search (or plant) any drugs or illegal substances within your home. The best way to avoid an invasion is to stand in the doorway of your front door. Do not step outside or go inside. If they ask to come in just state that you are comfortable where you are. ALWAYS ask if they have a search warrant. If they don't you can stash or trash anything you might not want to be found while they are out getting a warrant.

lol...lol ... they could try... as to whether they succeeded thats a different story

Again, under almost all circumstances, the police need a warrant to enter your house. If they ask if they can just come inside for such reasons as to eleminate you as a suspect, and you say yes, they are in. The police cannot even stay on your property if you say no and ask them to leave. However, please know, all of the laws are usually in favor of the local police and you have to be very carefull.

In many countries they cannot enter your home without a search warrant,as well as in the USA.