Let's 1st understand what these planets represent on their own?

Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.

Moon - Moon represents Mind, Emotions, Mother, Local Government, Peace of Mind, Home Environment, Water, Milk etc.

Rahu - It represents Illusion, Foreign Lands, Foreign Things (foreign as in unknown things. So, for someone living in Punjab, culture and rituals of Kerala is also unknown and hence Foreign), Drugs, Medicines, any type of intoxication, Fame, Wealth, Success, Obsession, Past-life incomplete Desires, Movies, Television, Online World, Cheating, Imagined Fears, Unusual Things, Unique Things, Creativity, Rule breaker. Rahu also blows things out of proportions.

3 planetary conjunction - Whenever 3 or more planets are conjunct with each other in a house/sign then things related to that house/sign become a kind of life focus for person. It is because of huge accumulation of energy within 30 degrees of sky. It means not only these 3 planets' energy is going into this house/sign but the energy of at least 4 to 6 house/sign is going into this house/sign as these planets will be ruling some houses in chart. So, this house/sign with conjunction becomes almost the most important house in chart. During Career Consultation, I give a huge importance to any such house which has 3 or more planets in it.

House Ruler - Besides these karaka things, Sun, Moon & Rahu will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include things related to the houses at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas.

Meaning of Sun-Moon-Rahu Conjunction - We have yet another stressful conjunction here. This is a stressful conjunction for both Sun & Moon as they are conjunct with their enemy Rahu. If Moon is in between Sun & Rahu degree-wise, then it creates a kind of Paap Kartari Yoga as gentle Moon would be sitting in between 2 strong malefics. In such cases, mind will take lots of stress and can be termed as depressed mind or it may lead to emotional imbalance. 1st of all this conjunction in a birth-chart shows that person was born during eclipse month and close to lunar eclipse. As Sun-Moon both are impacted by Rahu, this may lead to a situation where person has no idea about his real parents and may be brought up by foster parents. Even if situations with parents are not messed up to this level still there will be some illusion regarding them. If Sun or Moon are exalted, then person can have strength to face the hard impacts of Rahu but even then Rahu can have its malefic impact on person's ego and emotions. Such a person may not be able to balance his emotions and may have emotional issues from mother's side. They can also have ego related issues or ego conflicts due to closeness of Rahu with Sun. They can abuse power and authority. Just like any other conjunction with strong malefic, this conjunction will give positive results after mid-30s or after maturity age of Rahu, i.e. 42 years. They can have a very illusive or unique personality. At the same time, this conjunction can bring great fame. 

Who will win the Conjunction? - Rahu, till the end of 30s as it is impacting Sun & Moon both. 

Dignity & Strength - 

In any sign, Rahu will play its role as stated above. So, dignity doesn't really matter here.