"Ramen noodles" are a predominantly North American term for what the Japanese call "Chinese Noodles" (Chukamen, which I've also seen spelled Yuukamen).
In practice, you can use any wheat noodle that's made with eggs or kansui, including lamian or mee pok, or even buckwheat noodles (notably soba). Noodles made without either of those are not recommended.
It does not matter which of the above you use, although the Chukamen varieties are the more traditional. Much like pasta, it's largely a matter of personal preferences and how you want it to come out in terms of texture and appearance.
You'll find a wide variety of all of these noodles in dried or fresh form at any Asian supermarket or grocery store. Across Canada there's a chain called T&T. You should have no trouble finding some Asian grocery stores in your region with a phone book or maybe Google or Yelp.
The short answer is yes, they can come in a curly form. There are several types of fresh noodle used in Rāmen, which can be classified mainly according to thickness and shape.
Noodles are classified in shape into the straight sutorēto-men (ストレート麺), the curly chijire-men(縮れ面), and the more rare flat hirauchi-men(平打ち麺) . With the exception of the flat type, these are further divided into several degrees of thickness: the extra thick gokubuto-men(極太麺), the thick futo-men(太麺), the middle thick chūbuto-men(中太麺), the middle thin chūboso-men(中細麺), the thin hoso-men(細麺), and the extra thin gokuboso-men(極細麺). The thin types are also further differentiated in the amount of water they have in their dough.
Which shape and thickness of noodle is used depends on the style of ramen and on the chef's preference. For example, hakata rāmen(博多ラーメン), of the sorts served by Ippūdō in their Japanese stores, usually uses straight gokuboso type, but the same franchise uses chijire gokuboso type for their spicy rāmen and chijire gokubuto for their tsukemen. Likewise, yokohama tonkotsu-shōyu (横浜豚骨醤油), also known as ie-kei(家系), uses chijire-gokubuto.
The choice of noodle is related to many factors, but cooking time is an important one. In hakata rāmen, where the practice of topping up your noodles (kaedama/替え玉) is common, one has to be able to cook the noodle quickly, and hence the very thin noodle. Thicker types of noodle need longer cooking time (aprox. 10 min for gokubuto-men), and are only used by rāmen traditions that do not offer a noodle refill.
Another factor affecting the noodle type is the type of broth. Thicker broths ask for straight type of noodles, and thinner broths for curly noodles.
These are not rules etched in stone, however, as rāmen itself is a sort of creole food born from a mix of Japanese and Chinese cuisines. Some would say that there is not right or wrong in rāmen, and that anything is possible. However, what I have described above is what is customary and what you will find if you go to rāmen shops in Japan.
By the way, not all rāmen dishes in Japan use fresh noodles. There is a type of rāmen that originates from Chiba prefecture and is called takeoka rāmen(竹岡ラーメン), which was originally cooked by part-timer old ladies hired from the neighbourhood, and uses dried (what you would call instant) noodles because they are easier to cook.
Best Answer
Here are two really good, really easy ways to make ramen with eggs:
The easiest way is to just boil an egg and toss it in. If you eat this a lot, you can boil a few and keep them in the fridge. They should stay good for about 3-5 days. If you like liquid yolk in your soup, soft boil the eggs (boiled between about 3 and 9 minutes, to taste). Just cook your ramen, peel the egg, and drop it in for about 90 seconds right before you eat it. Cut the egg in half if you like. It is best to use older eggs for this. What I like to do personally (and remember this comes from the internet, so listen to me at your own risk) is use them for boiling if they have passed the expiration date, since that date (in the US) is based on the thickness of the egg white, not food safety. Older eggs don't stick to the shell when they are boiled, but new eggs do.
The slightly harder (but tastier) way is to poach the egg in the ramen. To do that, cook your ramen, and make sure you have tasted your broth and you like it. If you don't like the seasoning pack from the ramen, you can add any combination of soy sauce, grated ginger, crushed garlic, oyster sauce, miso paste, sriracha, etc. There are lots of tasty, cheap things to add depending on where you live. Try them! Anyway, back to the eggs... You want to cook the ramen until is is soft but not cooked all the way. Turn the heat down to almost nothing. Crack a very fresh egg (fresh eggs have a nice thick white) into the pan once it stops bubbling. Give it a few minutes. If you want to feel busy, you can spoon the hot broth onto the top of the egg to cook it faster. If you are particular about how cooked it is, jab a knife into the middle of the yolk and check it out.
Here is the last egg and ramen tip: My friends and I just called this noodle slop, but it tastes good, uses fresh at least a few fresh ingredients, and is super cheap. Cook some ramen noodles and drain them. In a big skillet, add some oil and fry some combination of celery, onions, chopped cabbage, ginger, fresh chili peppers, garlic, matchstick carrots, bell peppers, chopped scallions, cucumber, zucchini, etc. You want to cut them to the size you get in Chinese takeout and cook them until they get a bit soft. Then, scramble one egg per ramen package in with the veggies. When the egg is cooked, add the noodles and mix it all together. Add some salt and pepper, or the seasoning from the ramen if you are into the whole MSG thing. It is like cheap chow mein.