The question title is about "Canonical examples of algebraic structures", but the text of the question (and most answers) make(s) me think I can probably answer in more generality to: "What do you do in front of a new (set of) definition(s)?", or "What do you take when a theorem asks to consider ...?".

I generally first check with too trivial examples, before I go to trivial but probably more interesting examples (as found in other answers) ; and in a third pass, I even consider objects which don't have one of the wanted properties (counter-examples, also discussed in other anwers). So let me tell more about that first step, since it was overlooked so far.

Of course, the defect is that for some situations you end up with an example so simple that it doesn't give anything. But those same correct but too primitive examples can later turn up as interesting and enlightening counter-examples in other definitions/results (I give two such situations below the list), and that gives them some merit.

Here is a short list of witless examples, to demonstrate how weak those first examples can/should be:

• A set: $\emptyset$ ;
• A monoid: {0} ;
• A group: {0} ;
• A ring: {0} ;
• A vector space: {0} ;
• An algebra: {0} ;
• An element in a structure : the zero or unit, if it fits the bill (if it doesn't, its time will come in the third step)
• A morphism: the zero map, if it fits the bill (if it doesn't... see above) or the identity ;
• A symmetry in an object: the identity (or -id) ;
• A projection in an object: the identity or the zero map ;
• A graded something: either grade with the zero ring or $\mathbb{Z}/2\mathbb{Z}$ ;
• A ring of integers in a number field: $\mathbb{Z}$ ;
• (not 100% algebraic) A distance on a space: the one which is zero on a diagonal pair and one outside ;
• (the same idea, but in a more algebraic setting) A valuation on a field: the trivial one ;

Here is a first example of the usefulness of being thickheaded: the definition of integral ring excludes the zero ring, and the definition of a prime element in a ring excludes the units. Both are linked to the question of the {0} ring, through the definition of a prime ideal by giving an integral quotient.

Now for a second such situation: finding an example of distance which doesn't come from a norm is trivial if your very first idea of a distance is the foolish one ; the same question is hard if you take honest examples from the second step, where you'll probably only have the usual suspects on $\mathbb{R}^2$.