What influences do Confucianism, Taoism and Legalism have in modern China?

They have no relevance in modern China. They have been replaced by a Marxist interpretation of society and the state.

G= They still strongly influence the culture of the people in their acceptance of speedy execution for crimes we would not consider capital, Tai’Chi and Gung Fu, along with Shao Lin’s expanding influence are taught by Taoist centers and widely used. The educated people are expected to behave in a Confucian manner.
60 years of Communist rule that is increasingly driven by money rights
have overshadowed even Buddhism and the three you mentioned. They have not caused them to disappear.

During the cultural revolution in the 60s, these 3 have been suppress by the red guards.
However, Deng Xiao Peng realize that these 3 along with Buddhism are what makes a chinese – chinese. Most educated people cannot make a proper sentence without adding a few famous quotes from these 4 philosophies in their speech.
After Deng Xiao Peng took power and open up the country, the chinese people also realize that these are their cultural heritage so they are making a come back.

Confucianism has Always influenced China especially in intellectual.The influence can be found through the way the top leaders talked.
Taoism had begun to decay for several huge archaeological findings in recent 30 years.
Legalism had little influence in modern China.