Cannot index property 'displayName' of type String into type List of Variant

Generally that error means that you are trying to index a field that doesn't exist, so it often occurs when your variable is null or empty for some reason. There are two main ways to address this:

• Determine why your data isn't returning data for those fields (and whether it is valid or not).
• If it's valid that your field may not be present in the JSON result, then use the property or index function to use error handling. Here's an example:

property(
  local!Json.tags,
  "displayName",
  {}
)

Some other general suggestions - the expression that you have provided above is neither valid JSON nor a valid Appian expression, so it may be contributing to why you aren't getting the correct result. Also, I don't see why you actually need to count the display name - couldn't you just count the total number of tags there are?

[
{
"description": ".",
"displayName": ""
},
{
"description": "",
"displayName": ""
},
{
"description": ".",
"displayName": ""
}
]

this is my json. how to assign this to localvariable and access the count and property in foreach.

Just use a!fromJson() to convert from a JSON value to an Appian value and then use property + count on the result to get what you need.

a!fromJson("[{
  ""transId"":9223372036854775807,
  ""submitTimeUTC"":""2019-07-16T14:52:26Z"",
  ""submitTimeLocal"":""2019-07-16T07:52:26"",
  ""transactionOk"":true,
  ""accountType"":""Visa"",
  ""accountNumber"":123456,
  ""settleAmount"":412.97,
  ""product"":"""",
  ""marketType"":[""Business"",""Clothing"",""Child""]
}]")

i took the above code as reference. the json text in the above example has 2 double quotes.
the actual json which i am using has the property with in single double quotes. do i need to add anything here?

a!fromJson("[{
  ""transId"":9223372036854775807,
  ""submitTimeUTC"":""2019-07-16T14:52:26Z"",
  ""submitTimeLocal"":""2019-07-16T07:52:26"",
  ""transactionOk"":true,
  ""accountType"":""Visa"",
  ""accountNumber"":123456,
  ""settleAmount"":412.97,
  ""product"":"""",
  ""marketType"":[""Business"",""Clothing"",""Child""]
}]")

i took the above code as reference. the json text in the above example has 2 double quotes.
the actual json which i am using has the property with in single double quotes. do i need to add anything here?

While working with Appian expressions, you might see a different number of double quotes because the expression editor will not consider the code as plain text rather it will consider it as an expression. 

A better way to understand this is - Create a variable of type text and then use it inside a!fromJson() function. 
Now you can pass your JSON as a test value in the variable. 

pls share some samples

Sure, here's an example of what Harshit is saying - if you use a rule input and provide the full JSON in the input, it will ensure that the quotations are set up correctly.

count(
  property(
    property(
      a!fromJson(ri!json), 
      "tags", 
      {}
    ),
    "description",
    {}
  )
)

Thanks for the response. This will be helpful.