There are two questions here, and a couple of assumptions. I'm going to try to answer your question, but I need to tease out the assumptions so that I don't offer a misleading assumption.
(Prologue - for reasons that will become apparent, I suspect that BenCole's answer will be better than mine.)
The first "What are the commonly practiced variations of ikkyo?" is unfortunately probably impossible to answer. I suspect that the ". . . most commonly practiced. . ." portion of the question probably has meaning only within the context of your dojo (or conceivably of your Sensei). I can't verify that because I haven't trained Aikikai - I defer to BenCole on that. To the best of my knowledge, the only lineage that could answer "most common" is Tomiki, because it is the only lineage that has a formal curriculum. When I've watched Aikikai (and Iwama), the techniques practiced have been dependent on the Sensei's agenda for the day. I had the distinct sense that he presented one technique that fascinated him, then proceeded to demonstrate variations on that technique that would help his students to explore the portions of the techniques that were giving them trouble. For example, start with simple ikkyo, then show ikkyo tenshin, then because the students were having trouble with tenshin, move to another tenshin technique. If the students demonstrate adequate tenshin, try a technique that highlights sen-sen-no-sen (Quite possibly not a term used in Aikikai; it has to do with initiative, and if you don't have it, tenshin Ikkyo will be tougher.) etc. If I'm correct about that, then "the most commonly practiced Ikkyo" is the Ikkyo that students are having the most trouble with (or have the most opportunity to learn from.)
The second question has to do with the structure of aikido attacks - the "grammar" if you will. Bencole has given a table which illustrates many of the variations (I'll comment there on a few more possibilities, and suggest it may be community wiki) and at there is a decent glossary but Aikido is kind of like a lego set. You can assemble the modules in many different ways (See ShuHaRi for a fairly advanced explanation of the concept. You can choose an entry, and then a direction (omote/Ura), and then a control, and then a throw. That is the key to Bencole's table. Stenudd Sensei's book is a good reference for the attack variations. There are loosely four controls - Ikkyo, Nikyo, Sankyo and Yonkyo (Tomiki doesn't use those words, so I defer to somone with greater knowledge). But even that is misleading. In Tenkai Kote Hineri I can think of three or four finishes, which are not named as part of the throw. All are valid finishes. Two of them will proceed to a pin, and I can think of at least two different ways to do the pin - again, none of them are named. By definition, Shihonage has at least four different potential finishes to the throw, but there are at least two different entries, both of which go by the name Shihonage. A third and final example, I'm currently preparing to demonstrate for Nidan. Because my partner has trouble remembering, we built this table Techniques in the Koryu Dai San no Kata. You'll notice significant repetition in that table; very different techniques referred to by the same name.
So let me try to unify those two paragraphs(and rely on someone else to adapt the answer to Aikikai assumptions). The names are only rough indices to the techniques; they don't follow a formal grammar. Inside the dojo we use far more informal names (and I've been told that this is also somewhat true in Japan, although that is a different question). Kote Mwashi within our dojo is frequently known as "Kneel before Zod!" because it is easier to remember.
I think the goal underlying your question is a good one. But I want to warn you away from the assumption that there is a formal grammar.
Sorry, I've run on too long.