What are common ways for contemporary Christians to fast?

In Orthodox Christianity, fasting is vegan (no meat, no dairy, no egg, basically nothing of animal origin). And no alcohol (that means avoiding vinegar, as well).

Monks and nuns (and some of the more fervent Christians) do it the proper, old way - vegan and without using oil (like olive or sunflower oil, for example).

Another aspect to take in consideration is avoiding a lot of food - one can fall in gluttony with vegan food as well, and that's not fasting anymore.

But more important than the food fasting is the fasting for the soul:

• don't eat human flesh (don't argue with people, don't think bad of people, don't judge people, forgive everyone, be merciful towards everyone)
• fasting with our eyes (turn your head the other way when you see an attractive person)
• fasting with our legs (avoid going to sinful places, where we know we can fall)
• fast with your mind (keep your mind in continuous prayer; in Orthodoxy, this is done with the Jesus Prayer - "Lord Jesus Christ Son of God, have mercy on me a sinner", said without interruption in the mind)
• more prayer than usual, more meditation on the sufferings of our Lord
• meditate on the Final Judgement questions (Matthew 25:31-46) - how do we help the ones in need? There are a few billions of people out there without food, water, shelter etc

In the Catholic Church there are only two days of fasting in modern times:

Ash Wednesday and Good Friday are obligatory days of fasting and abstinence for Catholics. In addition, Fridays during Lent are obligatory days of abstinence.

For members of the Latin Catholic Church, the norms on fasting are obligatory from age 18 until age 59. When fasting, a person is permitted to eat one full meal, as well as two smaller meals that together are not equal to a full meal. The norms concerning abstinence from meat are binding upon members of the Latin Catholic Church from age 14 onwards.

Members of the Eastern Catholic Churches are to observe the particular law of their own sui iuris Church

If possible, the fast on Good Friday is continued until the Easter Vigil (on Holy Saturday night) as the "paschal fast" to honor the suffering and death of the Lord Jesus, and to prepare ourselves to share more fully and to celebrate more readily his Resurrection. - Fast & Abstinence

But that said, the Church allows us add to this in either number or adding different elements to our fast. I know some people who still fast the entire Lenten fast and others who do it on say on all Fridays of Lent or even on all Wednesdays and Fridays. Whatever one adds to the prescribed fast of the Church, please do so that one's health gets damaged. I also have a few friends who have a secondary list of things they give up in Lent in the case that something of their primary list is not an option, say as when one is at a business luncheon or meeting.

Add more – As Catholic men, we should never be satisfied with the bare minimum. We should seek to constantly pursue a deeper conversion. St. Francis de Sales gives some good advice in this regard:

If you are able to fast, you will do well to observe some days beyond what are ordered by the Church, for besides the ordinary effect of fasting in raising the mind, subduing the flesh, confirming goodness, and obtaining a heavenly reward, it is also a great matter to be able to control greediness, and to keep the sensual appetites and the whole body subject to the law of the Spirit; and although we may be able to do but little, the enemy nevertheless stands more in awe of those whom he knows can fast.

Accordingly, once you’ve begun to follow the law of the Church, build on that foundation to include fasting in other ways. Here are some ideas:

•Skip one meal extra a week, like breakfast or lunch. In addition to Fridays, Wednesdays are traditional days of fasting, so that might be a good day to start with.

•Deny yourself dessert on set days. Most of us eat too much sugar anyway.

•Skip salt on your food.

•Fast from soda. It’s terrible for you!

•Skip the beer or other alcoholic drinks when going out to eat.

•Don’t eat between meals. This sounds easy, but try it. You’ll find it’s quite hard since most of us snack frequently and don’t even realize it.

•Include things besides food. For example, fast from all technology one day a week.

•Fast (one main meal with two small snacks) one day a week.

•Drink only water. - Spiritual Weapons: Fasting