One of the most common examples of illogical reasoning is the straw man argument, which often entails either isolating a particular part of an argument before then presenting it as an entirely individual or separate position, or using a very early form of a theory to support a case while ignoring the fully developed theory itself (attacking Darwinism using only Darwin's work as a source for example, ignores all the progress made since, so is not valid). Finding a source that offers an extreme view or unrealistic account of a position, then treating it as fact, is also a straw man argument
Another form of illogical reasoning is the circular argument. This involves making a conclusion from an initial premise which is in turn entirely dependent on the conclusion itself, thus actually failing to prove anything. Circular arguments can be simple or complex, and can be harder to identify the more complex they become.
One example of a circular argument might be: Mr. smith is a great public speaker, because Mr. Smith has a knack of speaking well to people. This fails to prove Mr. Smith is a good public speaker at all, because the premise that he is a good speaker is based on the conclusion that he is a good speaker.