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**Do Muslims visit Our Lady of Walsingham's shrine?**
Shrines in Islam are prohibited, any Muslim who build a shrine OR put someone between him and Allah is doing a sin according to Sunni Islam, and visiting it is also not recommended, here is prohibition from Quran and hadith:
(Quran 7:194):
“Verily those whom ye call upon besides Allah are servants like unto you: Call upon them, and let them listen to your prayer, if ye are (indeed) truthful!”
إِنَّ الَّذِينَ تَدْعُونَ مِنْ دُونِ اللَّهِ عِبَادٌ أَمْثَالُكُمْ فَادْعُوهُمْ فَلْيَسْتَجِيبُوا لَكُمْ إِنْ كُنْتُمْ صَادِقِينَ
(Quran 39:43):
"What! Do they take for intercessors others besides Allah? Say: “Even if they have no power whatever and no intelligence?”
أَمِ اتَّخَذُوا مِنْ دُونِ اللَّهِ شُفَعَاءَ قُلْ أَوَلَوْ كَانُوا لا يَمْلِكُونَ شَيْئاً وَلا يَعْقِلُونَ
(Quran 17:56-57):
Say: Cry unto those (saints and angels) whom ye assume (to be gods) beside Him, yet they have no power to rid you of misfortune nor to change. Those unto whom they cry seek the way of approach to their Lord, which of them shall be the nearest; they hope for His mercy and they fear His doom. Lo! the doom of thy Lord is to be shunned.
قُلِ ادْعُوا الَّذِينَ زَعَمْتُمْ مِنْ دُونِهِ فَلَا يَمْلِكُونَ كَشْفَ الضُّرِّ عَنْكُمْ وَلَا تَحْوِيلًا أُولَئِكَ الَّذِينَ يَدْعُونَ يَبْتَغُونَ إِلَى رَبِّهِمُ الْوَسِيلَةَ أَيُّهُمْ أَقْرَبُ وَيَرْجُونَ رَحْمَتَهُ وَيَخَافُونَ عَذَابَهُ إِنَّ عَذَابَ رَبِّكَ كَانَ مَحْذُورًا
Sahih Muslim, Book 004, Number 1079:
'A’isha reported: The Messenger of Allah (may peace be upon him) said during his illness from which he never recovered: Allah cursed the Jews and the Christians that they took the graves of their prophets as mosques. She ('A’isha) reported: Had it not been so, his (Prophet’s) grave would have been in an open place, but it could not be due to the fear that it may not be taken as a mosque.