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Thanks, everyone, for your continued feedback. Here is the latest revision.

(defun read-side (side) (format t "Enter a value for side ~d: ~%" side) (read))

(defun triangle-p (a b c) (and (< a (+ b c)) (< b (+ a c)) (< c (+ a b))))

(defun equilateral-p (a b c) (= a b c))

(defun isoceles-p (a b c) (or (= a b) (= b c)))

(defun right-triangle-p (a b c) 
    (destructuring-bind (long-side short-side-1 short-side-2) (sort (list a b c) #'>)
          (= (expt long-side 2) (+ (expt short-side-1 2) (expt short-side-2 2)))))

(defun status-message (message a b c) (format t "The shape ~d,~d,~d ~a ~%" a b c message))

(defun classify-triangle (a b c)
    (cond ((not (triangle-p a b c)) nil)
          ((right-triangle-p a b c) 'right)
              ((equilateral-p a b c) 'equilateral)
              ((isoceles-p a b c) 'isoceles)
                  (t 'triangle)))

(let* ((a (read-side 1)) (b (read-side 2)) (c (read-side 3)) 
    (triangle-type (classify-triangle a b c))
    (article (if (member triangle-type '(isoceles equilateral)) "an" "a"))
    (tri-name (if (eq triangle-type 'triangle) "non-right, non-isoceles, non-equilateral" triangle-type)))
    (format t "The shape (~d, ~d, ~d) is ~:[not a triangle~;~a ~(~a~) triangle~]." a b c triangle-type article tri-name))

After reading through your suggestion on my sorting question and then reading up on 'format', I tried to write a format statement to simplify the logic at the bottom of the code. I couldn't figure out how to do a format conditional based on a symbol, so instead I defined the triangle types as global variables with 'defineparameter' (is there a better way to define constants?). I also used the format loop which you described in your response to the my-sort-3 question to shorten the expression. The new code is as follows:

(defparameter *right* 0)
(defparameter *equilateral* 1)
(defparameter *isoceles* 2)
(defparameter *triangle* 3)
(defparameter *non-triangle* 4)
(defun read-side (side) (format t "Enter a value for side ~d: ~%" side) (read))

(defun triangle-p (a b c) (and (< a (+ b c)) (< b (+ a c)) (< c (+ a b))))

(defun equilateral-p (a b c) (= a b c))

(defun isoceles-p (a b c) (or (= a b) (= b c) (= a c)))

(defun right-triangle-p (a b c) 
  (destructuring-bind (long-side &rest short-sides) (sort (list a b c) #'>)
    (= (expt long-side 2) (+ (expt (first short-sides) 2) (expt (second short-sides) 2)))))

(defun status-message (message a b c) (format t "The shape ~d,~d,~d ~a ~%" a b c message))

(defun classify-triangle (a b c)
  (cond ((not (triangle-p a b c)) *non-triangle*)
    ((right-triangle-p a b c) *right*)
    ((equilateral-p a b c) *equilateral*)
    ((isoceles-p a b c) *isoceles*)
    (t *triangle*)))

(let* ((a (read-side 1)) (b (read-side 2)) (c (read-side 3)) (triangle-type (classify-triangle a b c)))
  (format t "classify triangle says ~a ~%" triangle-type)
  (format t "The shape (~{~d~^,  ~}) is ~[a right~;an equilateral~;an isoceles~;a non-right, non-isoceles, non-equilateral~;not a~] triangle." (list a b c) triangle-type))

Here is the latest revision:

(defun read-side (side) (format t "Enter a value for side ~d: ~%" side) (read))

(defun triangle-p (a b c) (and (< a (+ b c)) (< b (+ a c)) (< c (+ a b))))

(defun equilateral-p (a b c) (= a b c))

(defun isoceles-p (a b c) (or (= a b) (= b c)))

(defun right-triangle-p (a b c) 
  (destructuring-bind (long-side &rest short-sides) (sort (list a b c) #'>)
    (= (expt long-side 2) (+ (expt (first short-sides) 2) (expt (second short-sides) 2)))))

(defun status-message (message a b c) (format t "The shape ~d,~d,~d ~a ~%" a b c message))

(defun classify-triangle (a b c)
  (cond ((not (triangle-p a b c)))
    ((right-triangle-p a b c) 'right)
    ((equilateral-p a b c) 'equilateral)
    ((isoceles-p a b c) 'isoceles)
    (t 'triangle)))

(let* ((a (read-side 1)) (b (read-side 2)) (c (read-side 3)) (triangle-type (classify-triangle a b c)))
  (format t "classify triangle says ~a ~%" triangle-type)
  (cond ((eq 'right triangle-type) (status-message "is a right triangle." a b c))
    ((eq 'equilateral triangle-type) (status-message "is an equilateral triangle." a b c))
    ((eq 'isoceles triangle-type) (status-message "is an isoceles triangle." a b c))
    ((eq 'triangle triangle-type) (status-message "is a non-right, non-isoceles, non-equilateral triangle." a b c))
    (t (status-message "is not a triangle." a b c))))