user-1: Is the conventional self "conventional" in the same way as dharmas are?
So if the conventionality of dharmas means that they arise and disappear each moment, or that they don't exist from their own side, or that they leave no trace of themselves, or that they always have parts (etc., etc.) does the same apply to the conventional self?
I understand that both lack "substance": but does that mean the same thing here?
I'm interested in anyone having said "no".

user471: Regarding self as a convention:
If you sit down, you have a lap. When you stand up, where is the lap? Where has it gone to?
What a "lap" is, is just a convention, just a like a "fist".
Similarly, a being (satta), a person (puggala), a chariot (ratha) are all conventions. See SN 5.10.
The self (atta) can be seen in two forms.
In the first case, when am unenlightened person clings to it, it's simply a mental idea (Snp 4.14) in the mind that one clings to.
In the second case, when an enlightened one says "me", "I", "we", "our", "mine", he is just using conventions (SN 1.25).
Even in the first case, it's just a convention too. Why? What you call "self" is somehow pegged to some other concept like your body or your mind or your personality.
So, imagine if you go to the beach and spend a lot of time and effort to build a sand castle. Then someone comes and smashes it. You feel angry and say "why did you smash my sand castle?" "You", "my" and "sand castle" are all conventions. The anger appeared because of clinging.

Regarding emptiness:
The concept of "substance" (svabhava) comes in the Madhyamaka philosophy of Mahayana. Madhyamaka states that everything is empty of substance, which means it is empty of the meaning that is given to it by the mind. It's not what you think it is.
Anatta is the notion that everything is not self. The body, the mind, the five aggregates, mental ideas etc. are all not self.
Anatta is a specialized form of emptiness (i.e. everything is empty of a self - SN 35.85), while the Madhyamaka emptiness is a more generalized form (i.e. everything is empty of the meaning given to it by the mind). Both types of emptiness are covered in MN 1. So, Theravada has both too, in fact.
The Madhyamaka emptiness is explained by the Pali term papanca. If papanca is the process of baking, then svabhava is the cake. Please see this answer.
In his commentary to Snp 4.14, Ven. Thanissaro explained that the objectification and classification (papanca) of the mental idea of the self vs. non-self objects, lead to craving, clinging and suffering. He also explained papanca in his commentary to MN 18.
A detailed explanation can be found in this answer.

user13375: 
OP: "Is the conventional self "conventional" in the same way as dharmas
are?"

In extent Mahayana traditions the selflessness of phenomena and the selflessness of persons is of the same nature and both are posited. They differ only in the object of focus. Further, it is considered critical to understand the selflessness of phenomena in order to achieve the soteriological ends of the Buddha.
In the Theravada, the selflessness of phenomena is not posited and it is treated by some as not germane to the soteriological ends of the Buddha while others insist that positing the selflessness of phenomena is actually a hindrance in that it denies the reality of the objectively existing world. However, it should be noted that some Theravada practitioners here on the site seem to have more or less the same understanding of the two selves as Mahayana.

OP: "So if the conventionality of dharmas means that they arise and
disappear each moment, or that they don't exist from their own side,
or that they leave no trace of themselves, or that they always have
parts (etc., etc.) does the same apply to the conventional self?"

In Mahayana: yes. In Theravada: see above.

OP: "I understand that both lack "substance": but does that mean the
same thing here?"

Again see above.
Also, these questions and answers might be helpful.

-1: <p>Is the conventional self &quot;conventional&quot; in the same way as dharmas are?</p>
<p>So if the conventionality of dharmas means that they arise and disappear each moment, or that they don't exist from their own side, or that they leave no trace of themselves, or that they always have parts (etc., etc.) does the same apply to the conventional self?</p>
<p>I understand that both lack &quot;substance&quot;: but does that mean the same thing here?</p>
<p><strong>I'm interested in anyone having said &quot;no&quot;.</strong></p>
 471: <p>Regarding self as a convention:</p>
<p>If you sit down, you have a lap. When you stand up, where is the lap? Where has it gone to?</p>
<p>What a &quot;lap&quot; is, is just a convention, just a like a &quot;fist&quot;.</p>
<p>Similarly, a being (<em>satta</em>), a person (<em>puggala</em>), a chariot (<em>ratha</em>) are all conventions. See <a href="https://suttacentral.net/sn5.10/en/bodhi" rel="nofollow noreferrer">SN 5.10</a>.</p>
<p>The self (<em>atta</em>) can be seen in two forms.</p>
<p>In the first case, when am unenlightened person clings to it, it's simply a mental idea (<a href="https://www.accesstoinsight.org/tipitaka/kn/snp/snp.4.14.than.html" rel="nofollow noreferrer">Snp 4.14</a>) in the mind that one clings to.</p>
<p>In the second case, when an enlightened one says &quot;me&quot;, &quot;I&quot;, &quot;we&quot;, &quot;our&quot;, &quot;mine&quot;, he is just using conventions (<a href="https://suttacentral.net/sn1.25/en/sujato" rel="nofollow noreferrer">SN 1.25</a>).</p>
<p>Even in the first case, it's just a convention too. Why? What you call &quot;self&quot; is somehow pegged to some other concept like your body or your mind or your personality.</p>
<p>So, imagine if you go to the beach and spend a lot of time and effort to build a sand castle. Then someone comes and smashes it. You feel angry and say &quot;why did you smash my sand castle?&quot; &quot;You&quot;, &quot;my&quot; and &quot;sand castle&quot; are all conventions. The anger appeared because of clinging.</p>
<hr />
<p>Regarding emptiness:</p>
<p>The concept of &quot;substance&quot; (<em>svabhava</em>) comes in the Madhyamaka philosophy of Mahayana. Madhyamaka states that everything is empty of substance, which means it is empty of the meaning that is given to it by the mind. It's not what you think it is.</p>
<p><em>Anatta</em> is the notion that everything is not self. The body, the mind, the five aggregates, mental ideas etc. are all not self.</p>
<p><em>Anatta</em> is a specialized form of emptiness (i.e. everything is empty of a self - <a href="https://www.accesstoinsight.org/tipitaka/sn/sn35/sn35.085.than.html" rel="nofollow noreferrer">SN 35.85</a>), while the Madhyamaka emptiness is a more generalized form (i.e. everything is empty of the meaning given to it by the mind). Both types of emptiness are covered in <a href="https://suttacentral.net/mn1/en/bodhi" rel="nofollow noreferrer">MN 1</a>. So, Theravada has both too, in fact.</p>
<p>The Madhyamaka emptiness is explained by the Pali term <em>papanca</em>. If <em>papanca</em> is the process of baking, then <em>svabhava</em> is the cake. Please see <a href="https://buddhism.stackexchange.com/a/39254/471">this answer</a>.</p>
<p>In <a href="https://www.accesstoinsight.org/tipitaka/kn/snp/snp.4.14.than.html#fn-1" rel="nofollow noreferrer">his commentary to Snp 4.14</a>, Ven. Thanissaro explained that the objectification and classification (<em>papanca</em>) of the mental idea of the self vs. non-self objects, lead to craving, clinging and suffering. He also explained <em>papanca</em> in <a href="https://www.accesstoinsight.org/tipitaka/mn/mn.018.than.html" rel="nofollow noreferrer">his commentary to MN 18</a>.</p>
<p>A detailed explanation can be found in <a href="https://buddhism.stackexchange.com/a/44535/471">this answer</a>.</p>
 13375: <blockquote>
<p>OP: &quot;Is the conventional self &quot;conventional&quot; in the same way as dharmas
are?&quot;</p>
</blockquote>
<p>In extent Mahayana traditions the selflessness of phenomena and the selflessness of persons is of the same nature and both are posited. They differ only in the object of focus. Further, it is considered critical to understand the selflessness of phenomena in order to achieve the soteriological ends of the Buddha.</p>
<p>In the Theravada, the selflessness of phenomena is not posited and it is treated by some as not germane to the soteriological ends of the Buddha while others insist that positing the selflessness of phenomena is actually a hindrance in that it denies the reality of the objectively existing world. However, it should be noted that <em>some</em> <a href="https://buddhism.stackexchange.com/a/45629/13375">Theravada practitioners here on the site</a> seem to have more or less the same understanding of the two selves as Mahayana.</p>
<blockquote>
<p>OP: &quot;So if the conventionality of dharmas means that they arise and
disappear each moment, or that they don't exist from their own side,
or that they leave no trace of themselves, or that they always have
parts (etc., etc.) does the same apply to the conventional self?&quot;</p>
</blockquote>
<p>In Mahayana: yes. In Theravada: see above.</p>
<blockquote>
<p>OP: &quot;I understand that both lack &quot;substance&quot;: but does that mean the
same thing here?&quot;</p>
</blockquote>
<p>Again see above.</p>
<p>Also, these <a href="https://buddhism.stackexchange.com/questions/28708/does-theravada-posit-the-selflessness-of-phenomena">questions</a> and <a href="https://buddhism.stackexchange.com/questions/28709/if-theravada-doesnt-posit-the-selflessness-of-phenomena-then-how-to-interpret">answers</a> might be helpful.</p>
