必须在配置中指定<input>的确切名称属性。"."和"_"不可互换。由于您的名称不是有效的标识符，因此需要用引号将其括起来。

rules: {
    "Report.Name": {
      required: true
    }
},
messages: {
    "Report.Name": {
      required: "Testing: Report_Name required"
    }
},

你也可以考虑使用vanilla JavaScript。如果表单应该只提交到某个URL，则可以省略最后一个事件侦听器（onsubmit）。
oninvalid事件侦听器（第二个）依赖于input元素上的required属性。当这个属性被设置时，输入将根据maxlength属性进行验证，并且还将假设该值不是空字符串。

// Any input to any input in the form
document.forms.form01.addEventListener('input', e => {
  // if some input, test if input is valid and if so
  // remove the class name 'invalid' being there or not
  if (e.target.validity.valid) {
    e.target.classList.remove('invalid');
  }
});

// If submit button if clicked or the function checkValidity()
// on the form is called this eventlistener will trigger.
// This overrides the default message by calling e.preventDefault();
document.forms.form01.addEventListener('invalid', e => {
  e.preventDefault();
  // when something is invalid (input element being e.target)
  // set the class name 'invalid'
  e.target.classList.add('invalid');
}, true);// notice the parameter 'true'

// If form is valide and the form is submitting
document.forms.form01.addEventListener('submit', e => {
  e.preventDefault();
  console.log('form submitting...');
}, false);

form {
  display: flex;
  flex-direction: column;
}

input~span {
  display: none;
  background-color: orange;
}

input.invalid~span {
  display: inline;
}

<form name="form01">
  <label>
    <span>Report name:</span>
    <input type="text" class="form-control"
      data-val-required="The Report Name field is required." 
      data-val="true" 
      data-val-maxlength="The field Report Name must be a string or array type with a maximum length of &#x27;35&#x27;." 
      data-val-maxlength-max="35"  
      maxlength="35" 
      name="Report.Name"
      required/>
    <span class="text-danger">The field Report Name must be a string or array type with a maximum length of &#x27;35&#x27;.</span>
  </label>
  <label>
    <button type="submit">Submit</button>
  </label>
</form>

不需要spedifying规则，消息，或消息。开箱即用的属性工作得很好。我最终需要的是“errorClass”来获得一个错误消息以显示在屏幕上。jQuery不是最好的文档化语言。

$("#frmSaveReport").validate({
    submitHandler: function (form) {
        submitSavedReport();
    },
    errorClass: "text-danger"
});