Something that is permanent, continuous, controllable, malleable to one's expectation, something that results only in pleasantness, etc. can be considered self. Unfortunately there isn't. Even Nirvana cannot be and should not be identified with self.

If 'Atta' is physical body?

Body is subjected to old age, decay and death hence not self. You leave your body when you die. At this stage you might think the mind is Atta.

If 'Atta' is mind (Citta)?

Even consciousness of the mind, which gives the illusion of an observer, is not continuous. See this answer for more details.

If rebirth is a footprint of 'Atta'?

This will seem so when when you do not see in consistency in the mind or part of the mind which is consciousness. When you see that the mind and consciousness is inconsistent and broken (there are gaps, i.e, there is a gap, it arises, stays, passes away, followed by a gap again) you see there is no self transferred. Current state arises due to the previous state in inputs (Āhāra) to the current state. So rebirth is not entirely different or completely new person, but since there is nothing that is permanent or consistent, nothing identify as absolutely pleasurable it is not self.

If wholesome acts is 'Atta'?

Fabrications is not self as they are not entirely pleasant. In fact all that is conditioned things end in unpleasantness. Hence the motivation to find Nirvana where there is no arising, birth hence no unpleasantness. This also should not be taken as self. If you do you create craving and Nirvana will more further away from you.

Wholesome acts bring pleasantness but they also come to an end which is unpleasant. In this sense it cannot be taken as self.

From the viewpoint of Buddhist practise, as explained in SN 22.81, 'atta' ('self') refers to an ignorance created 'thought formation' (sankhara) that ignorantly assumes self-identity & ownership in relation to any of the five aggregates.

This ignorant assumption of self-identity & ownership is central in the generation of suffering, as explained in SN 22.1 and as explained in Dependent Origination (SN 12.2).

Therefore, in Buddhism, the primary importance of 'atta' is it is a central cause of suffering that is ever-present in the mind of the unenlightened person ('uninstructed worldling').

Here, bhikkhus, the uninstructed worldling ... assumes form as self (attato). That assumption, bhikkhus, is a formation (saṅkhāro). That formation — what is its source, what is its origin, from what is it born (jatiko) and produced? When the uninstructed worldling is contacted by a feeling born of ignorance-contact, craving arises: thence that formation (of 'self') is born. SN 22.81

There is the case where an uninstructed, run-of-the-mill person — assumes form (the body) to be the self or the self as possessing form or form as in the self or the self as in form. He is seized with the idea that 'I am form' or 'Form is mine.' As he is seized with these ideas, his form changes & alters, and he falls into sorrow, lamentation, pain, distress & despair over its change & alteration.

There is the case where a well-instructed disciple of the noble ones — who has regard for noble ones, is well-versed & disciplined in their Dhamma; who has regard for men of integrity, is well-versed & disciplined in their Dhamma — does not assume form to be the self or the self as possessing form or form as in the self or the self as in form. He is not seized with the idea that 'I am form' or 'Form is mine.' As he is not seized with these ideas, his form changes & alters, but he does not fall into sorrow, lamentation, pain, distress or despair over its change & alteration.

SN 22.1

And what, bhikkhus, is attachment? There are these four kinds of attachment: attachment of sensual pleasures, attachment of views, attachment of rules and vows, attachment of words of 'self' (attavā­du­pādā­naṃ). This is called attachment.

And what, bhikkhus, is birth? The birth of the various beings (sattānaṃ) into the various orders of beings (sattanikāye)...

SN 12.2

Note: 'Satta' is a synonym for 'atta', in 'satta' is also a view about 'self'. Refer to SN 5.10 & SN 23.2.

Therefore, bhikkhus, any kind of form whatsoever ... Any kind of feeling whatsoever … Any kind of perception whatsoever … Any kind of volitional formations whatsoever … Any kind of consciousness whatsoever, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near, all consciousness should be seen as it really is with correct wisdom thus: ‘This is not mine (mama), this I am (asmi) not, this is not my self (atta).’ SN 22.59

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It is said that the world is empty, the world is empty, lord. In what respect is it said that the world is empty? Insofar as it is empty of a self (attena) or of anything pertaining to a self (attaniyena): Thus it is said, Ananda, that the world is empty Suñña Sutta