Cook potatoes in large pot of boiling salted water for 10 minutes. Add green beans and carrots; cook five minutes, or until potatoes are tender when pierced with fork. Drain; cool under cold running water.
Whisk all ingredients from olive down in a small bowl until well-blended.
When potatoes are cool enough to handle, quarter large ones and half small ones. Place in large bowl, along with beans, carrots, spinach, radishes, and green onions. Toss with enough vinaigrette to lightly coat. Serve at room temperature.